mmtheorems38 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3701-3800 - Page 38 of 107

Type	Label	Description
Statement
Theorem	f1oco 3701	Composition of one-to-one onto functions.
|- ((F:B-1-1-onto->C /\ G:A-1-1-onto->B) -> (F o. G):A-1-1-onto->C)
Theorem	f1ococnv2 3702	The composition of a one-to-one onto function and its converse equals the identity relation restricted to the function's range.
|- (F:A-1-1-onto->B -> (F o. `'F) = (I |` B))
Theorem	f1ococnv1 3703	The composition of a one-to-one onto function's converse and itself equals the identity relation restricted to the function's domain.
|- (F:A-1-1-onto->B -> (`'F o. F) = (I |` A))
Theorem	f1dmex 3704	If the codomain of a one-to-one function exists, so does its domain. This theorem is equivalent to the Axiom of Replacement ax-rep 2689.
|- ((F:A-1-1->B /\ B e. C) -> A e. V)
Theorem	ffoss 3705	Relationship between a mapping and an onto mapping. Figure 38 of [Enderton] p. 145.
|- F e. V   =>   |- (F:A-->B <-> E.x(F:A-onto->x /\ x (_ B))
Theorem	f11o 3706	Relationship between one-to-one and one-to-one onto function.
|- F e. V   =>   |- (F:A-1-1->B <-> E.x(F:A-1-1-onto->x /\ x (_ B))
Theorem	f10 3707	The empty set maps one-to-one into any class.
|- (/):(/)-1-1->A
Theorem	f1o00 3708	One-to-one onto mapping of the empty set.
|- (F:(/)-1-1-onto->A <-> (F = (/) /\ A = (/)))
Theorem	fo00 3709	Onto mapping of the empty set.
|- (F:(/)-onto->A <-> (F = (/) /\ A = (/)))
Theorem	f1o0 3710	One-to-one onto mapping of the empty set.
|- (/):(/)-1-1-onto->(/)
Theorem	f1oi 3711	A restriction of the identity relation is a one-to-one onto function.
|- (I |` A):A-1-1-onto->A
Theorem	f1ovi 3712	The identity relation is a one-to-one onto function on the universe.
|- I:V-1-1-onto->V
Theorem	f1osn 3713	A singleton of an ordered pair is one-to-one onto function.
|- A e. V   &   |- B e. V   =>   |- {<.A, B>.}:{A}-1-1-onto->{B}
Theorem	fv2 3714	Alternate definition of function value. Definition 10.11 of [Quine] p. 68.
|- A e. V   =>   |- (F` A) = U.{x | A.y(AFy <-> y = x)}
Theorem	fvprc 3715	A function's value at a proper class is the empty set.
|- (-. A e. V -> (F` A) = (/))
Theorem	elfv 3716	Membership in a function value.
|- B e. V   =>   |- (A e. (F` B) <-> E.x(A e. x /\ A.y(BFy <-> y = x)))
Theorem	fveq1 3717	Equality theorem for function value.
|- (F = G -> (F` A) = (G` A))
Theorem	fveq2 3718	Equality theorem for function value.
|- (A = B -> (F` A) = (F` B))
Theorem	fveq1i 3719	Equality inference for function value.
|- F = G   =>   |- (F` A) = (G` A)
Theorem	fveq1d 3720	Equality deduction for function value.
|- (ph -> F = G)   =>   |- (ph -> (F` A) = (G` A))
Theorem	fveq2i 3721	Equality inference for function value.
|- A = B   =>   |- (F` A) = (F` B)
Theorem	fveq2d 3722	Equality deduction for function value.
|- (ph -> A = B)   =>   |- (ph -> (F` A) = (F` B))
Theorem	hbfv 3723	Bound-variable hypothesis builder for function value.
|- (y e. F -> A.x y e. F)   &   |- (y e. A -> A.x y e. A)   =>   |- (y e. (F` A) -> A.x y e. (F` A))
Theorem	hbfvd 3724	Deduction version of bound-variable hypothesis builder hbfv 3723. If a closed theorem version is desired, see hbfvd2 3725.
|- (ph -> A.xph)   &   |- (ph -> (y e. F -> A.x y e. F))   &   |- (ph -> (y e. A -> A.x y e. A))   =>   |- (ph -> (y e. (F` A) -> A.x y e. (F` A)))
Theorem	hbfvd2 3725	Deduction version of bound-variable hypothesis builder hbfv 3723. This variant of hbfvd 3724 allows us to create a closed theorem form by replacing the uncommitted antecedent ph with an appropriate substitution instance.
|- (ph -> A.xA.yph)   &   |- (ph -> (y e. F -> A.x y e. F))   &   |- (ph -> (y e. A -> A.x y e. A))   =>   |- (ph -> (y e. (F` A) -> A.x y e. (F` A)))
Theorem	fvex 3726	The value of a class exists. Corollary 6.13 of [TakeutiZaring] p. 27.
|- (F` A) e. V
Theorem	fv3 3727	Alternate definition of the value of a function. Definition 6.11 of [TakeutiZaring] p. 26.
|- A e. V   =>   |- (F` A) = {x | (E.y(x e. y /\ AFy) /\ E!y AFy)}
Theorem	fvres 3728	The value of a restricted function.
|- (A e. B -> ((F |` B)` A) = (F` A))
Theorem	funssfv 3729	The value of a member of the domain of a subclass of a function.
|- ((Fun F /\ G (_ F /\ A e. dom G) -> (F` A) = (G` A))
Theorem	tz6.12-1 3730	Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27.
|- A e. V   =>   |- ((AFy /\ E!y AFy) -> (F` A) = y)
Theorem	tz6.12 3731	Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27.
|- A e. V   =>   |- ((<.A, y>. e. F /\ E!y<.A, y>. e. F) -> (F` A) = y)
Theorem	tz6.12f 3732	Function value, using bound-variable hypotheses instead of distinct variable conditions.
|- (w e. F -> A.y w e. F)   =>   |- ((<.x, y>. e. F /\ E!y<.x, y>. e. F) -> (F` x) = y)
Theorem	tz6.12-2 3733	Function value when F is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27.
|- (-. E!y AFy -> (F` A) = (/))
Theorem	tz6.12c 3734	Corollary of Theorem 6.12(1) of [TakeutiZaring] p. 27.
|- A e. V   =>   |- (E!y AFy -> ((F` A) = y <-> AFy))
Theorem	tz6.12i 3735	Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27.
|- A e. V   =>   |- (B =/= (/) -> ((F` A) = B -> AFB))
Theorem	csbfv12g 3736	Move class substitution in and out of a function value.
|- (A e. C -> [_A / x]_(F` B) = ([_A / x]_F` [_A / x]_B))
Theorem	csbfv2g 3737	Move class substitution in and out of a function value.
|- (A e. C -> [_A / x]_(F` B) = (F` [_A / x]_B))
Theorem	csbfvg 3738	Substitution for a function value.
|- (A e. C -> [_A / x]_(F` x) = (F` A))
Theorem	ndmfv 3739	The value of a class outside its domain is the empty set.
|- (-. A e. dom F -> (F` A) = (/))
Theorem	ndmfvrcl 3740	Reverse closure law for function with the empty set not in its domain.
|- dom F = S   &   |- -. (/) e. S   =>   |- ((F` A) e. S -> A e. S)
Theorem	elfvdm 3741	If a function value has a member, the argument belongs to the domain.
|- (A e. (F` B) -> B e. dom F)
Theorem	nfvres 3742	A non-element of a restriction has empty value.
|- (-. A e. B -> ((F |` B)` A) = (/))
Theorem	fveqres 3743	Equal values imply equal values in a restriction.
|- ((F` A) = (G` A) -> ((F |` B)` A) = ((G |` B)` A))
Theorem	funbrfv 3744	The second argument of a binary relation on a function is the function's value.
|- B e. V   =>   |- (Fun F -> (AFB -> (F` A) = B))
Theorem	funopfv 3745	The second element in an ordered pair member of a function is the function's value.
|- B e. V   =>   |- (Fun F -> (<.A, B>. e. F -> (F` A) = B))
Theorem	funopfvg 3746	The second element in an ordered pair member of a function is the function's value.
|- ((B e. C /\ Fun F) -> (<.A, B>. e. F -> (F` A) = B))
Theorem	fnbrfvb 3747	Equivalence of function value and binary relation.
|- C e. V   =>   |- ((F Fn A /\ B e. A) -> ((F` B) = C <-> BFC))
Theorem	fnopfvb 3748	Equivalence of function value and ordered pair membership.
|- C e. V   =>   |- ((F Fn A /\ B e. A) -> ((F` B) = C <-> <.B, C>. e. F))
Theorem	funbrfvb 3749	Equivalence of function value and binary relation.
|- B e. V   =>   |- ((Fun F /\ A e. dom F) -> ((F` A) = B <-> AFB))
Theorem	funopfvb 3750	Equivalence of function value and ordered pair membership. Theorem 4.3(ii) of [Monk1] p. 42.
|- B e. V   =>   |- ((Fun F /\ A e. dom F) -> ((F` A) = B <-> <.A, B>. e. F))
Theorem	funbrfvbg 3751	Function value in terms of a binary relation.
|- ((Fun F /\ A e. dom F /\ B e. C) -> ((F` A) = B <-> AFB))
Theorem	fnopabfv 3752	Representation of a function in terms of its values.
|- (F Fn A <-> F = {<.x, y>. | (x e. A /\ y = (F` x))})
Theorem	fnrnfv 3753	The range of a function expressed as a collection of the function's values.
|- (F Fn A -> ran F = {y | E.x e. A y = (F` x)})
Theorem	fvelrnb 3754	A member of a function's range is a value of the function.
|- (F Fn A -> (B e. ran F <-> E.x e. A (F` x) = B))
Theorem	dfimafn 3755	Alternate definition of the image of a function. (Contributed by Raph Levien, 20-Nov-2006.)
|- ((Fun F /\ A (_ dom F) -> (F"A) = {y | E.x e. A (F` x) = y})
Theorem	dfimafn2 3756	Alternate definition of the image of a function as an indexed union of singletons of function values. (Contributed by Raph Levien, 20-Nov-2006.)
|- ((Fun F /\ A (_ dom F) -> (F"A) = U_x e. A {(F` x)})
Theorem	funimass4 3757	Membership relation for the values of a function whose image is a subclass. (Contributed by Raph Levien, 20-Nov-2006.)
|- ((Fun F /\ A (_ dom F) -> ((F"A) (_ B <-> A.x e. A (F` x) e. B))
Theorem	fvelima 3758	Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42.
|- ((Fun F /\ A e. (F"B)) -> E.x e. B (F` x) = A)
Theorem	fvelimab 3759	Function value in an image.
|- ((F Fn A /\ B (_ A) -> (C e. (F"B) <-> E.x e. B (