Theorem fvopab3ig 3773

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 1532	. . . . . . . . 9 |- (x = A -> (x e. C <-> A e. C))
2		fvopab3ig.1	. . . . . . . . 9 |- (x = A -> (ph <-> ps))
3	1, 2	anbi12d 627	. . . . . . . 8 |- (x = A -> ((x e. C /\ ph) <-> (A e. C /\ ps)))
4		fvopab3ig.2	. . . . . . . . 9 |- (y = B -> (ps <-> ch))
5	4	anbi2d 615	. . . . . . . 8 |- (y = B -> ((A e. C /\ ps) <-> (A e. C /\ ch)))
6	3, 5	opelopabg 2813	. . . . . . 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 3544	. . . . . 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 1428	. . . . . . 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 987	. . . . 5 |- Fun {<.x, y>. | (x e. C /\ ph)}
16		funopfvg 3747	. . . . 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 695	. . . 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 3720	. . 3 |- (F` A) = ({<.x, y>. | (x e. C /\ ph)}` A)
22	21	eqeq1i 1480	. 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 955   e. wcel 957  E*wmo 1380  <.cop 2408  {copab 2662  Fun wfun 3172  ` cfv 3178

This theorem is referenced by:  fvopab4g 3774  oprabval6g 4027

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fv 3194

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