Theorem fvopab3ig 2869

Description: Value of a function given by ordered pair abstraction.

Hypotheses

Ref	Expression
fvopab3ig.1	|- (x = A -> (ph <-> ps))
fvopab3ig.2	|- (y = B -> (ps <-> ch))
fvopab3ig.3	|- (x e. C -> E*yph)
fvopab3ig.4	|- F = {<.x, y>. | (x e. C /\ ph)}

Assertion

Ref	Expression
fvopab3ig	|- ((A e. C /\ B e. D) -> (ch -> (F` A) = B))

Distinct variable group(s):   x,y,A   x,B,y   x,C,y   ch,x,y

Proof of Theorem fvopab3ig

Step	Hyp	Ref	Expression
1		eleq1 1149	. . . . . . . . 9 |- (x = A -> (x e. C <-> A e. C))
2		fvopab3ig.1	. . . . . . . . 9 |- (x = A -> (ph <-> ps))
3	1, 2	anbi12d 476	. . . . . . . 8 |- (x = A -> ((x e. C /\ ph) <-> (A e. C /\ ps)))
4		fvopab3ig.2	. . . . . . . . 9 |- (y = B -> (ps <-> ch))
5	4	anbi2d 468	. . . . . . . 8 |- (y = B -> ((A e. C /\ ps) <-> (A e. C /\ ch)))
6	3, 5	opelopabg 2115	. . . . . . 7 |- ((A e. C /\ B e. D) -> (<.A, B>. e. {<.x, y>. | (x e. C /\ ph)} <-> (A e. C /\ ch)))
7	6	biimpar 325	. . . . . 6 |- (((A e. C /\ B e. D) /\ (A e. C /\ ch)) -> <.A, B>. e. {<.x, y>. | (x e. C /\ ph)})
8	7	exp43 301	. . . . 5 |- (A e. C -> (B e. D -> (A e. C -> (ch -> <.A, B>. e. {<.x, y>. | (x e. C /\ ph)}))))
9	8	pm2.43a 60	. . . 4 |- (A e. C -> (B e. D -> (ch -> <.A, B>. e. {<.x, y>. | (x e. C /\ ph)})))
10	9	imp 277	. . 3 |- ((A e. C /\ B e. D) -> (ch -> <.A, B>. e. {<.x, y>. | (x e. C /\ ph)}))
11		funopab 2694	. . . . . 6 |- (Fun {<.x, y>. | (x e. C /\ ph)} <-> A.xE*y(x e. C /\ ph))
12		fvopab3ig.3	. . . . . . 7 |- (x e. C -> E*yph)
13		moanimv 1052	. . . . . . 7 |- (E*y(x e. C /\ ph) <-> (x e. C -> E*yph))
14	12, 13	mpbir 165	. . . . . 6 |- E*y(x e. C /\ ph)
15	11, 14	mpgbir 686	. . . . 5 |- Fun {<.x, y>. | (x e. C /\ ph)}
16		funopfvg 2854	. . . . 5 |- ((B e. D /\ Fun {<.x, y>. | (x e. C /\ ph)}) -> (<.A, B>. e. {<.x, y>. | (x e. C /\ ph)} -> ({<.x, y>. | (x e. C /\ ph)}` A) = B))
17	15, 16	mpan2 519	. . . 4 |- (B e. D -> (<.A, B>. e. {<.x, y>. | (x e. C /\ ph)} -> ({<.x, y>. | (x e. C /\ ph)}` A) = B))
18	17	adantl 305	. . 3 |- ((A e. C /\ B e. D) -> (<.A, B>. e. {<.x, y>. | (x e. C /\ ph)} -> ({<.x, y>. | (x e. C /\ ph)}` A) = B))
19	10, 18	syld 27	. 2 |- ((A e. C /\ B e. D) -> (ch -> ({<.x, y>. | (x e. C /\ ph)}` A) = B))
20		fvopab3ig.4	. . . 4 |- F = {<.x, y>. | (x e. C /\ ph)}
21	20	fveq1i 2833	. . 3 |- (F` A) = ({<.x, y>. | (x e. C /\ ph)}` A)
22	21	cleq1i 1108	. 2 |- ((F` A) = B <-> ({<.x, y>. | (x e. C /\ ph)}` A) = B)
23	19, 22	syl6ibr 186	1 |- ((A e. C /\ B e. D) -> (ch -> (F` A) = B))

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E*wmo 1008   = wceq 1091   e. wcel 1092  <.cop 1810  {copab 2055  Fun wfun 2416  ` cfv 2422

This theorem is referenced by:  fvopab4g 2870

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-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

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