mmtheorems41 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 4001-4100 - Page 41 of 108

Type	Label	Description
Statement
Theorem	reloprab 4001	An operation class abstraction is a relation.
|- Rel {<.<.x, y>., z>. | ph}
Theorem	hboprab1 4002	The abstraction variables in an operation class abstraction are not free.
|- (w e. {<.<.x, y>., z>. | ph} -> A.x w e. {<.<.x, y>., z>. | ph})
Theorem	hboprab2 4003	The abstraction variables in an operation class abstraction are not free.
|- (w e. {<.<.x, y>., z>. | ph} -> A.y w e. {<.<.x, y>., z>. | ph})
Theorem	oprabbid 4004	Equivalent wff's yield equal operation class abstractions (deduction rule).
|- (ph -> A.xph)   &   |- (ph -> A.yph)   &   |- (ph -> A.zph)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> {<.<.x, y>., z>. | ps} = {<.<.x, y>., z>. | ch})
Theorem	oprabbidv 4005	Equivalent wff's yield equal operation class abstractions (deduction rule).
|- (ph -> (ps <-> ch))   =>   |- (ph -> {<.<.x, y>., z>. | ps} = {<.<.x, y>., z>. | ch})
Theorem	oprabbii 4006	Equivalent wff's yield equal operation class abstractions.
|- (ph <-> ps)   =>   |- {<.<.x, y>., z>. | ph} = {<.<.x, y>., z>. | ps}
Theorem	cbvoprab12 4007	Rule used to change first two bound variables in an operation abstraction, using implicit substitution.
|- (ph -> A.wph)   &   |- (ph -> A.vph)   &   |- (ps -> A.xps)   &   |- (ps -> A.yps)   &   |- ((x = w /\ y = v) -> (ph <-> ps))   =>   |- {<.<.x, y>., z>. | ph} = {<.<.w, v>., z>. | ps}
Theorem	cbvoprab12v 4008	Rule used to change first two bound variables in an operation abstraction, using implicit substitution.
|- ((x = w /\ y = v) -> (ph <-> ps))   =>   |- {<.<.x, y>., z>. | ph} = {<.<.w, v>., z>. | ps}
Theorem	cbvoprab3v 4009	Rule used to change the third bound variable in an operation abstraction, using implicit substitution.
|- (z = w -> (ph <-> ps))   =>   |- {<.<.x, y>., z>. | ph} = {<.<.x, y>., w>. | ps}
Theorem	elimdeloprv 4010	Eliminate a hypothesis which is a predicate expressing membership in the result of an operator (deduction version). See ghomgrplem 10398 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)
|- (ph -> C e. (AFB))   &   |- Z e. (XFY)   =>   |- if(ph, C, Z) e. (if(ph, A, X)Fif(ph, B, Y))
Theorem	dmoprab 4011	The domain of an operation class abstraction.
|- dom {<.<.x, y>., z>. | ph} = {<.x, y>. | E.zph}
Theorem	dmoprabss 4012	The domain of an operation class abstraction.
|- dom {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ ph)} (_ (A X. B)
Theorem	rnoprab 4013	The range of an operation class abstraction.
|- ran {<.<.x, y>., z>. | ph} = {z | E.xE.yph}
Theorem	reldmoprab 4014	The domain of an operation class abstraction is a relation.
|- Rel dom {<.<.x, y>., z>. | ph}
Theorem	oprabss 4015	Structure of an operation class abstraction.
|- {<.<.x, y>., z>. | ph} (_ ((V X. V) X. V)
Theorem	eloprabg 4016	The law of concretion for operation class abstraction. Compare elopab 2820.
|- (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 4017	Inference of operation class abstraction subclass from implication.
|- (ph -> ps)   =>   |- {<.<.x, y>., z>. | ph} (_ {<.<.x, y>., z>. | ps}
Theorem	resoprab 4018	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 4019	"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 4020	"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 4021	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 4022	Functionality and domain of an operation class abstraction.
|- (ph -> E!zps)   =>   |- {<.<.x, y>., z>. | (ph /\ ps)} Fn {<.x, y>. | ph}
Theorem	ffnoprval 4023	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 4024	Closure law for an operation.
|- F:(R X. S)-->C   =>   |- ((A e. R /\ B e. S) -> (AFB) e. C)
Theorem	eqfnoprval 4025	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 4026	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 4027	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 4028	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 4029	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 4030	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 4031	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 4032	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 4033	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 4034	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 4035	The value of an operation class abstraction. A version of