Theorem fvopab3ig 3778

Description: Value of a function given by ordered-pair class 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 groups:   x,y,A   x,B,y   x,C,y   ch,x,y

Proof of Theorem fvopab3ig

Step	Hyp	Ref	Expression
1		eleq1 1534	. . . . . . . . 9 |- (x = A -> (x e. C <-> A e. C))
2		fvopab3ig.1	. . . . . . . . 9 |- (x = A -> (ph <-> ps))
3	1, 2	anbi12d 628	. . . . . . . 8 |- (x = A -> ((x e. C /\ ph) <-> (A e. C /\ ps)))
4		fvopab3ig.2	. . . . . . . . 9 |- (y = B -> (ps <-> ch))
5	4	anbi2d 616	. . . . . . . 8 |- (y = B -> ((A e. C /\ ps) <-> (A e. C /\ ch)))
6	3, 5	opelopabg 2817	. . . . . . 7 |- ((A e. C /\ B e. D) -> (<.A, B>. e. {<.x, y>. | (x e. C /\ ph)} <-> (A e. C /\ ch)))
7	6	biimpar 417	. . . . . 6 |- (((A e. C /\ B e. D) /\ (A e. C /\ ch)) -> <.A, B>. e. {<.x, y>. | (x e. C /\ ph)})
8	7	exp43 384	. . . . 5 |- (A e. C -> (B e. D -> (A e. C -> (ch -> <.A, B>. e. {<.x, y>. | (x e. C /\ ph)}))))
9	8	pm2.43a 66	. . . 4 |- (A e. C -> (B e. D -> (ch -> <.A, B>. e. {<.x, y>. | (x e. C /\ ph)})))
10	9	imp 350	. . 3 |- ((A e. C /\ B e. D) -> (ch -> <.A, B>. e. {<.x, y>. | (x e. C /\ ph)}))
11		funopab 3548	. . . . . 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 1429	. . . . . . 7 |- (E*y(x e. C /\ ph) <-> (x e. C -> E*yph))
14	12, 13	mpbir 190	. . . . . 6 |- E*y(x e. C /\ ph)
15	11, 14	mpgbir 988	. . . . 5 |- Fun {<.x, y>. | (x e. C /\ ph)}
16		funopfvg 3752	. . . . 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 696	. . . 4 |- (B e. D -> (<.A, B>. e. {<.x, y>. | (x e. C /\ ph)} -> ({<.x, y>. | (x e. C /\ ph)}` A) = B))
18	17	adantl 388	. . 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 3725	. . 3 |- (F` A) = ({<.x, y>. | (x e. C /\ ph)}` A)
22	21	eqeq1i 1482	. 2 |- ((F` A) = B <-> ({<.x, y>. | (x e. C /\ ph)}` A) = B)
23	19, 22	syl6ibr 213	1 |- ((A e. C /\ B e. D) -> (ch -> (F` A) = B))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E*wmo 1381  <.cop 2411  {copab 2666  Fun wfun 3176  ` cfv 3182

This theorem is referenced by:  fvopab4g 3779  oprabval6g 4032

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198

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