Theorem oprabvalig 4009

Description: The value of an operation class 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 groups:   x,y,z,A   x,B,y,z   x,C,y,z   x,R,y,z   x,S,y,z   ps,x   ch,x,y   th,x,y,z

Proof of Theorem oprabvalig

Step	Hyp	Ref	Expression
1		eleq1 1526	. . . . . . . . . . 11 |- (x = A -> (x e. R <-> A e. R))
2	1	anbi1d 615	. . . . . . . . . 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 626	. . . . . . . . 9 |- (x = A -> (((x e. R /\ y e. S) /\ ph) <-> ((A e. R /\ y e. S) /\ ps)))
5		eleq1 1526	. . . . . . . . . . 11 |- (y = B -> (y e. S <-> B e. S))
6	5	anbi2d 614	. . . . . . . . . 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 626	. . . . . . . . 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 614	. . . . . . . . 9 |- (z = C -> (((A e. R /\ B e. S) /\ ch) <-> ((A e. R /\ B e. S) /\ th)))
11	4, 8, 10	eloprabg 3992	. . . . . . . 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 417	. . . . . . 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 377	. . . . . 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 11	. . . . 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 797	. . . 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 64	. . 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	. . . . . . 7 |- ((x e. R /\ y e. S) -> E*zph)
18		moanimv 1422	. . . . . . 7 |- (E*z((x e. R /\ y e. S) /\ ph) <-> ((x e. R /\ y e. S) -> E*zph))
19	17, 18	mpbir 190	. . . . . 6 |- E*z((x e. R /\ y e. S) /\ ph)
20	19	funoprab 3996	. . . . 5 |- Fun {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
21		funopfvg 3737	. . . . 5 |- ((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 694	. . . 4 |- (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	3ad2ant3 800	. . 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))
24	16, 23	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))
25		df-opr 3950	. . . 4 |- (AFB) = (F` <.A, B>.)
26		oprabvalig.5	. . . . 5 |- F = {<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}
27	26	fveq1i 3710	. . . 4 |- (F` <.A, B>.) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
28	25, 27	eqtr 1487	. . 3 |- (AFB) = ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.)
29	28	eqeq1i 1474	. 2 |- ((AFB) = C <-> ({<.<.x, y>., z>. | ((x e. R /\ y e. S) /\ ph)}` <.A, B>.) = C)
30	24, 29	syl6ibr 213	1 |- ((A e. R /\ B e. S /\ C e. D) -> (th -> (AFB) = C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  E*wmo 1374  <.cop 2401  Fun wfun 3166  ` cfv 3172  (class class class)co 3948  {copab2 3949

This theorem is referenced by:  oprabvali 4010  oprabval2gf 4011  spwval2 8577

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fv 3188  df-opr 3950  df-oprab 3951

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