mmtheorems41 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4001-4100 - Page 41 of 107

Type	Label	Description
Statement
Theorem	eloprabg 4001	The law of concretion for operation class abstraction. Compare elopab 2807.
|- (x = A -> (ph <-> ps))   &   |- (y = B -> (ps <-> ch))   &   |- (z = C -> (ch <-> th))   =>   |- ((A e. D /\ B e. R /\ C e. S) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ph} <-> th))
Theorem	ssoprab2i 4002	Inference of operation class abstraction subclass from implication.
|- (ph -> ps)   =>   |- {<.<.x, y>., z>. | ph} (_ {<.<.x, y>., z>. | ps}
Theorem	resoprab 4003	Restriction of an operation class abstraction.
|- ({<.<.x, y>., z>. | ph} |` (A X. B)) = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ ph)}
Theorem	funoprabg 4004	"At most one" is a sufficient condition for an operation class abstraction to be a function.
|- (A.xA.yE*zph -> Fun {<.<.x, y>., z>. | ph})
Theorem	funoprab 4005	"At most one" is a sufficient condition for an operation class abstraction to be a function.
|- E*zph   =>   |- Fun {<.<.x, y>., z>. | ph}
Theorem	fnoprabg 4006	Functionality and domain of an operation class abstraction.
|- (A.xA.y(ph -> E!zps) -> {<.<.x, y>., z>. | (ph /\ ps)} Fn {<.x, y>. | ph})
Theorem	fnoprab 4007	Functionality and domain of an operation class abstraction.
|- (ph -> E!zps)   =>   |- {<.<.x, y>., z>. | (ph /\ ps)} Fn {<.x, y>. | ph}
Theorem	ffnoprval 4008	An operation maps to a class to which all values belong.
|- (F:(A X. B)-->C <-> (F Fn (A X. B) /\ A.x e. A A.y e. B (xFy) e. C))
Theorem	foprcl 4009	Closure law for an operation.
|- F:(R X. S)-->C   =>   |- ((A e. R /\ B e. S) -> (AFB) e. C)
Theorem	eqfnoprval 4010	Equality of two operations is determined by their values.
|- ((F Fn (A X. B) /\ G Fn (C X. D)) -> (F = G <-> ((A X. B) = (C X. D) /\ A.x e. A A.y e. B (xFy) = (xGy))))
Theorem	fnoprval 4011	Representation of an operation class abstraction in terms of its values.
|- (F Fn (A X. B) <-> F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = (xFy))})
Theorem	foprval 4012	Representation of an operation class abstraction in terms of its values.
|- (F:(A X. B)-->C <-> (F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = (xFy))} /\ A.x e. A A.y e. B (xFy) e. C))
Theorem	oprabex 4013	Existence of an operation class abstraction.
|- A e. V   &   |- B e. V   &   |- ((x e. A /\ y e. B) -> E*zph)   &   |- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ ph)}   =>   |- F e. V
Theorem	oprabex2g 4014	Existence of an operation class abstraction (special case).
|- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}   =>   |- ((A e. R /\ B e. S) -> F e. V)
Theorem	oprabex2 4015	Existence of an operation class abstraction (special case).
|- A e. V   &   |- B e. V   &   |- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}   =>   |- F e. V
Theorem	oprabex3 4016	Existence of an operation class abstraction (special case).
|- H e. V   &   |- F = {<.<.x, y>., z>. | ((x e. (H X. H) /\ y e. (H X. H)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R))}   =>   |- F e. V
Theorem	oprabval 4017	The value of an operation class abstraction.
|- C e. V   &   |- (x = A -> (ph <-> ps))   &   |- (y = B -> (ps <-> ch))   &   |- (z = C -> (ch <-> th))   &   |- ((x e. R /\ y e. S) -> E!zph)   &   |- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}   =>   |- ((A e. R /\ B e. S) -> ((AFB) = C <-> th))
Theorem	oprabvalig 4018	The value of an operation class abstraction (weak version).
|- (x = A -> (ph <-> ps))   &   |- (y = B -> (ps <-> ch))   &   |- (z = C -> (ch <-> th))   &   |- ((x e. R /\ y e. S) -> E*zph)   &   |- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}   =>   |- ((A e. R /\ B e. S /\ C e. D) -> (th -> (AFB) = C))
Theorem	oprabvali 4019	The value of an operation class abstraction (weak version).
|- C e. V   &   |- (x = A -> (ph <-> ps))   &   |- (y = B -> (ps <-> ch))   &   |- (z = C -> (ch <-> th))   &   |- ((x e. R /\ y e. S) -> E*zph)   &   |- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}   =>   |- ((A e. R /\ B e. S) -> (th -> (AFB) = C))
Theorem	oprabval2gf 4020	The value of an operation class abstraction. A version of oprabval2g 4021 using bound-variable hypotheses.
|- (w e. G -> A.x w e. G)   &   |- (w e. S -> A.y w e. S)   &   |- (x = A -> R = G)   &   |- (y = B -> G = S)   &   |- F = {<.<.x, y>., z>. | ((x e. C /\ y e. D) /\ z = R)}   =>   |- ((A e. C /\ B e. D /\ S e. H) -> (AFB) = S)
Theorem	oprabval2g 4021	The value of an operation class abstraction. Special case.
|- (x = A -> R = G)   &   |- (y = B -> G = S)   &   |- F = {<.<.x, y>., z>. | ((x e. C /\ y e. D) /\ z = R)}   =>   |- ((A e. C /\ B e. D /\ S e. H) -> (AFB) = S)
Theorem	oprabval2 4022	The value of an operation class abstraction. Special case.
|- S e. V   &   |- (x = A -> R = G)   &   |- (y = B -> G = S)   &   |- F = {<.<.x, y>., z>. | ((x e. C /\ y e. D) /\ z = R)}   =>   |- ((A e. C /\ B e. D) -> (AFB) = S)
Theorem	oprabval5 4023	The value of an operation class abstraction. Special case.
|- S e. V   &   |- (x = A -> R = G)   &   |- (y = B -> G = S)   &   |- F = {<.<.x, y>., z>. | z = R}   =>   |- ((A e. C /\ B e. D) -> (AFB) = S)
Theorem	oprabval3 4024	The value of an operation class abstraction. Special case.
|- S e. V   &   |- (((w = A /\ v = B) /\ (u = C /\ f = D)) -> R = S)   &   |- F = {<.<.x, y>., z>. | ((x e. (H X. H) /\ y e. (H X. H)) /\ E.wE.vE.uE.f((x = <.w, v>. /\ y = <.u, f>.) /\ z = R))}   =>   |- (((A e. H /\ B e. H) /\ (C e. H /\ D e. H)) -> (<.A, B>.F<.C, D>.) = S)
Theorem	oprabval4g 4025	Value of an operation given by an ordered-pair class abstraction. (This is the operation analog of fvopab2 3785.)
|- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}   =>   |- ((x e. A /\ y e. B /\ C e. D) -> (xFy) = C)
Theorem	oprabval6g 4026	The value of an operation class abstraction. Special case.
|- (<.x, y>. = <.A, B>. -> R = S)   &   |- F = {<.<.x, y>., z>. | (<.x, y>. e. C /\ z = R)}   =>   |- (((A e. G /\ B e. H /\ <.A, B>. e. C) /\ S e. J) -> (AFB) = S)
Theorem	oprvalres 4027	The value of a restricted operation. (Contributed by FL, 10-Nov-2006.)
|- ((A e. C /\ B e. D) -> (A(F |` (C X. D))B) = (AFB))
Theorem	oprssoprval 4028	The value of a member of the domain of a subclass of an operation.
|- (((Fun F /\ G Fn