Theorem oprabvalig 3048

Description: The value of an operation abstraction (weak version).

Hypotheses

Ref	Expression
oprabvalig.1	|- (x = A -> (ph <-> ps))
oprabvalig.2	|- (y = B -> (ps <-> ch))
oprabvalig.3	|- (z = C -> (ch <-> th))
oprabvalig.4	|- ((x e. R /\ y e. S) -> E*zph)
oprabvalig.5	|- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}

Assertion

Ref	Expression
oprabvalig	|- ((A e. R /\ B e. S /\ C e. D) -> (th -> (AFB) = C))

Distinct variable group(s):   x,y,z,A   x,B,y,z   x,C,y,z   x,R,y,z   x,S,y,z   x,D,y,z   ps,x   ch,x,y   th,x,y,z

Proof of Theorem oprabvalig

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . . . . . . . 11 |- (x = A -> (x e. R <-> A e. R))
2	1	anbi1d 469	. . . . . . . . . 10 |- (x = A -> ((x e. R /\ y e. S) <-> (A e. R /\ y e. S)))
3		oprabvalig.1	. . . . . . . . . 10 |- (x = A -> (ph <-> ps))
4	2, 3	anbi12d 476	. . . . . . . . 9 |- (x = A -> (((x e. R /\ y e. S) /\ ph) <-> ((A e. R /\ y e. S) /\ ps)))
5		eleq1 1149	. . . . . . . . . . 11 |- (y = B -> (y e. S <-> B e. S))
6	5	anbi2d 468	. . . . . . . . . 10 |- (y = B -> ((A e. R /\ y e. S) <-> (A e. R /\ B e. S)))
7		oprabvalig.2	. . . . . . . . . 10 |- (y = B -> (ps <-> ch))
8	6, 7	anbi12d 476	. . . . . . . . 9 |- (y = B -> (((A e. R /\ y e. S) /\ ps) <-> ((A e. R /\ B e. S) /\ ch)))
9		oprabvalig.3	. . . . . . . . . 10 |- (z = C -> (ch <-> th))
10	9	anbi2d 468	. . . . . . . . 9 |- (z = C -> (((A e. R /\ B e. S) /\ ch) <-> ((A e. R /\ B e. S) /\ th)))
11	4, 8, 10	eloprabg 3035	. . . . . . . 8 |- ((A e. R /\ B e. S /\ C e. D) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} <-> ((A e. R /\ B e. S) /\ th)))
12	11	biimpar 325	. . . . . . 7 |- (((A e. R /\ B e. S /\ C e. D) /\ ((A e. R /\ B e. S) /\ th)) -> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)})
13	12	exp32 294	. . . . . 6 |- ((A e. R /\ B e. S /\ C e. D) -> ((A e. R /\ B e. S) -> (th -> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)})))
14	13	com12 13	. . . . 5 |- ((A e. R /\ B e. S) -> ((A e. R /\ B e. S /\ C e. D) -> (th -> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)})))
15	14	3adant3 599	. . . 4 |- ((A e. R /\ B e. S /\ C e. D) -> ((A e. R /\ B e. S /\ C e. D) -> (th -> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)})))
16	15	pm2.43i 58	. . 3 |- ((A e. R /\ B e. S /\ C e. D) -> (th -> <.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}))
17		oprabvalig.4	. . . . . . . 8 |- ((x e. R /\ y e. S) -> E*zph)
18		moanimv 1052	. . . . . . . 8 |- (E*z((x e. R /\ y e. S) /\ ph) <-> ((x e. R /\ y e. S) -> E*zph))
19	17, 18	mpbir 165	. . . . . . 7 |- E*z((x e. R /\ y e. S) /\ ph)
20	19	funoprab 3037	. . . . . 6 |- Fun {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
21		funopfvg 2854	. . . . . 6 |- ((C e. D /\ Fun {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} -> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C))
22	20, 21	mpan2 519	. . . . 5 |- (C e. D -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} -> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C))
23	22	adantl 305	. . . 4 |- ((B e. S /\ C e. D) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} -> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C))
24	23	3adant1 597	. . 3 |- ((A e. R /\ B e. S /\ C e. D) -> (<.<.A, B>., C>. e. {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)} -> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C))
25	16, 24	syld 27	. 2 |- ((A e. R /\ B e. S /\ C e. D) -> (th -> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C))
26		df-opr 3003	. . . 4 |- (AFB) = (F` <.A, B>.)
27		oprabvalig.5	. . . . 5 |- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
28	27	fveq1i 2833	. . . 4 |- (F` <.A, B>.) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
29	26, 28	eqtr 1119	. . 3 |- (AFB) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
30	29	cleq1i 1108	. 2 |- ((AFB) = C <-> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C)
31	25, 30	syl6ibr 186	1 |- ((A e. R /\ B e. S /\ C e. D) -> (th -> (AFB) = C))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196   /\ w3a 581  E*wmo 1008   = wceq 1091   e. wcel 1092  <.cop 1810  Fun wfun 2416  ` cfv 2422  (class class class)co 3001  {copab2 3002

This theorem is referenced by:  oprabvali 3049  oprabval2g 3050

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fv 2438  df-opr 3003  df-oprab 3004

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