mmtheorems39 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 3801-3900 - Page 39 of 107

Type	Label	Description
Statement
Theorem	elrnopabg 3801	Membership in the range of an ordered pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ y = B)}   =>   |- (A.x e. A B e. D -> (C e. ran F <-> E.x e. A C = B))
Theorem	elrnopab 3802	Membership in the range of an ordered pair class abstraction.
|- B e. V   &   |- F = {<.x, y>. | (x e. A /\ y = B)}   =>   |- (C e. ran F <-> E.x e. A C = B)
Theorem	chfnrn 3803	The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain.
|- ((F Fn A /\ A.x e. A (F` x) e. x) -> ran F (_ U.A)
Theorem	funfvop 3804	Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41.
|- ((Fun F /\ A e. dom F) -> <.A, (F` A)>. e. F)
Theorem	fvimacnvi 3805	A member of a preimage is a function value argument.
|- ((Fun F /\ A e. (`'F"B)) -> (F` A) e. B)
Theorem	fvimacnv 3806	The argument of a function value belongs to the pre-image of any class containing the function value. (Contributed by Raph Levien, 20-Nov-2006. He remarks: "This proof is unsatisfying, because it seems to me that funimass2 3575 could probably be strengthened to a biconditional.")
|- ((Fun F /\ A e. dom F) -> ((F` A) e. B <-> A e. (`'F"B)))
Theorem	funimass3 3807	A kind of contraposition law that infers an image subclass from a subclass of a pre-image. (Contributed by Raph Levien, 20-Nov-2006. He remarks: "Likely this could be proved directly, and fvimacnv 3806 would be the special case of A being a singleton, but it works this way round too.")
|- ((Fun F /\ A (_ dom F) -> ((F"A) (_ B <-> A (_ (`'F"B)))
Theorem	funimass5 3808	A subclass of a preimage in terms of function values.
|- ((Fun F /\ A (_ dom F) -> (A (_ (`'F"B) <-> A.x e. A (F` x) e. B))
Theorem	funconstss 3809	Two ways of specifying that a function is constant on a subdomain.
|- ((Fun F /\ A (_ dom F) -> (A.x e. A (F` x) = B <-> A (_ (`'F"{B})))
Theorem	fvimacnvALT 3810	Another proof of fvimacnv 3806, based on funimass3 3807. If funimass3 3807 is ever proved directly, as opposed to using funimacnv 3573 pointwise, then the proof of funimacnv 3573 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.)
|- ((Fun F /\ A e. dom F) -> ((F` A) e. B <-> A e. (`'F"B)))
Theorem	fimacnv 3811	The pre-image of the codomain of a mapping is the mapping's domain. (Contributed by FL, 25-Jan-2007.)
|- (F:A-->B -> (`'F"B) = A)
Theorem	fnopfv 3812	Ordered pair with function value. Part of Theorem 4.3(i) of [Monk1] p. 41.
|- ((F Fn A /\ B e. A) -> <.B, (F` B)>. e. F)
Theorem	fvelrn 3813	A function's value belongs to its range.
|- ((Fun F /\ A e. dom F) -> (F` A) e. ran F)
Theorem	fnfvelrn 3814	A function's value belongs to its range.
|- ((F Fn A /\ B e. A) -> (F` B) e. ran F)
Theorem	ffvelrn 3815	A function's value belongs to its codomain.
|- ((F:A-->B /\ C e. A) -> (F` C) e. B)
Theorem	ffvelrni 3816	A function's value belongs to its codomain.
|- F:A-->B   =>   |- (C e. A -> (F` C) e. B)
Theorem	dff4 3817	Alternate definition of a mapping.
|- (F:A-->B <-> (F Fn A /\ F (_ (A X. B)))
Theorem	dff2 3818	Alternate definition of a mapping.
|- (F:A-->B <-> (F (_ (A X. B) /\ A.x e. A E!y xFy))
Theorem	dff3 3819	Alternate definition of a mapping.
|- (F:A-->B <-> (F (_ (A X. B) /\ A.x e. A E!y e. B xFy))
Theorem	dffo3 3820	An onto mapping expressed in terms of function values.
|- (F:A-onto->B <-> (F:A-->B /\ A.y e. B E.x e. A y = (F` x)))
Theorem	dffo4 3821	Alternate definition of an onto mapping.
|- (F:A-onto->B <-> (F:A-->B /\ A.y e. B E.x e. A xFy))
Theorem	dffo5 3822	Alternate definition of an onto mapping.
|- (F:A-onto->B <-> (F:A-->B /\ A.y e. B E.x xFy))
Theorem	exfo 3823	A relation equivalent to the existence of an onto mapping. The right-hand f is not necessarily a function.
|- (E.f f:A-onto->B <-> E.f(A.x e. A E!y e. B xfy /\ A.x e. B E.y e. A yfx))
Theorem	fopab2 3824	Functionality of an ordered-pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ y = C)}   =>   |- (A.x e. A C e. B <-> F:A-->B)
Theorem	fopabssxp 3825	Inclusion of a function in a cross product.
|- F = {<.x, y>. | (x e. A /\ y = C)}   =>   |- (A.x e. A C e. B -> F (_ (A X. B))
Theorem	rnssopab 3826	Range of a function that is expressed as an ordered-pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ y = C)}   &   |- C e. V   =>   |- (A.x e. A C e. B <-> ran F (_ B)
Theorem	fopab3 3827	Functionality of an ordered-pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ y = C)}   &   |- C e. V   =>   |- (ran F (_ B <-> F:A-->B)
Theorem	fopab 3828	Functionality of an ordered-pair class abstraction.
|- F = {<.x, y>. | (x e. A /\ y = C)}   &   |- (x e. A -> C e. B)   =>   |- F:A-->B
Theorem	ffnfv 3829	A function maps to a class to which all values belong.
|- (F:A-->B <-> (F Fn A /\ A.x e. A (F` x) e. B))
Theorem	ffnfvf 3830	A function maps to a class to which all values belong. This version of ffnfv 3829 uses bound-variable hypotheses instead of distinct variable conditions.
|- (y e. A -> A.x y e. A)   &   |- (y e. B -> A.x y e. B)   &   |- (y e. F -> A.x y e. F)   =>   |- (F:A-->B <-> (F Fn A /\ A.x e. A (F` x) e. B))
Theorem	fnfvrnss 3831	An upper bound for range determined by function values.
|- ((F Fn A /\ A.x e. A (F` x) e. B) -> ran F (_ B)
Theorem	fopabfv 3832	Representation of a mapping in terms of its values.
|- (F:A-->B <-> (F = {<.x, y>. | (x e. A /\ y = (F` x))} /\ A.x e. A (F` x) e. B))
Theorem	fopabco 3833	Composition of two functions expressed as ordered-pair class abstractions. Note that v may be assigned to w, y, or z if desired.
|- R e. V   &   |- S e. V   &   |- T e. V   &   |- (z = R -> S = T)   &   |- F = {<.x, y>. | (x e. A /\ y = R)}   &   |- G = {<.z, w>. | (z e. B /\ w = S)}   &   |- H = {<.x, v>. | (x e. A /\ v = T)}   =>   |- (ran F (_ B -> (G o. F) = H)
Theorem	fopabcos 3834	Composition of two functions expressed as ordered-pair class abstractions.
|- C e. V   &   |- D e. V   &   |- F = {<.x, y>. | (x e. A /\ y = C)}   &   |- G = {<.x, y>. | (x e. B /\ y = D)}   =>   |- (ran G (_ A -> (F o. G) = {<.x, y>. | (x e. B /\ y = [_D / x]_C)})
Theorem	fsn 3835	A function maps a singleton to a singleton iff it is the singleton of a ordered pair.
|- A e. V   &   |- B e. V   =>   |- (F:{A}-->{B} <-> F = {<.A, B>.})
Theorem	xpsn 3836	The cross product of two singletons.
|- A e. V   &   |- B e. V   =>   |- ({A} X. {B}) = {<.A, B>.}
Theorem	fsn2 3837	A function that maps a singleton to a class is the singleton of an ordered pair.
|- A e. V   =>   |- (F:{A}-->B <-> ((F` A) e. B /\ F = {<.A, (F` A)>.}))
Theorem	fnressn 3838	A function restricted to a singleton.
|- ((F Fn A /\ B e. A) -> (F |` {B}) = {<.B, (F` B)>.})
Theorem	fressnfv 3839	The value of a function restricted to a singleton.
|- ((F Fn A /\ B e. A) -> ((F |` {B}):{B}-->C <-> (F` B) e. C))
Theorem	fvconst 3840	The value of a constant function.
|- ((F:A-->{B} /\ C e. A) -> (F` C) = B)
Theorem	fopabsn 3841	The singleton of an ordered pair expressed as an ordered pair class abstraction.
|- A e. V   &   |- B e. V   =>   |- {<.A, B>.} = {<.x, y>. | (x e. {A} /\ y = B)}
Theorem	fopabap 3842	Append an additional value to a function.
|- A e. V   &   |- B e. V   &   |- (R u. {A}) = S   &   |- (x = A -> C = B)   =>   |- ({<.x, y>. | (x e. R /\ y = C)} u. {<.A, B>.}) = {<.x, y>. | (x e. S /\ y = C)}
Theorem	fvi 3843	The value of the identity function.
|- (A e. B -> (I` A) = A)
Theorem	fvresi 3844	The value of a restricted identity function.
|- (B e. A -> ((I |` A)` B) = B)
Theorem	fvconst2g 3845	The value of a constant function.
|- ((B e. D /\ C e. A) -> ((A X. {B})` C) = B)
Theorem	fconst2g 3846	A constant function expressed as a cross product.
|- (B e. C -> (F:A-->{B} <-> F = (A X. {B})))
Theorem	fvconst2 3847	The value of a constant function.
|- B e. V   =>   |- (C e. A -> (