Theorem fvopab3ig 3854

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 1571	. . . . . . . . 9 |- (x = A -> (x e. C <-> A e. C))
2		fvopab3ig.1	. . . . . . . . 9 |- (x = A -> (ph <-> ps))
3	1, 2	anbi12d 630	. . . . . . . 8 |- (x = A -> ((x e. C /\ ph) <-> (A e. C /\ ps)))
4		fvopab3ig.2	. . . . . . . . 9 |- (y = B -> (ps <-> ch))
5	4	anbi2d 618	. . . . . . . 8 |- (y = B -> ((A e. C /\ ps) <-> (A e. C /\ ch)))
6	3, 5	opelopabg 2870	. . . . . . 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 348	. . 3 |- ((A e. C /\ B e. D) -> (ch -> <.A, B>. e. {<.x, y>. | (x e. C /\ ph)}))
11		funopab 3623	. . . . . 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 1462	. . . . . . 7 |- (E*y(x e. C /\ ph) <-> (x e. C -> E*yph))
14	12, 13	mpbir 188	. . . . . 6 |- E*y(x e. C /\ ph)
15	11, 14	mpgbir 1020	. . . . 5 |- Fun {<.x, y>. | (x e. C /\ ph)}
16		funopfvg 3828	. . . . 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 699	. . . 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 3801	. . 3 |- (F` A) = ({<.x, y>. | (x e. C /\ ph)}` A)
22	21	eqeq1i 1519	. 2 |- ((F` A) = B <-> ({<.x, y>. | (x e. C /\ ph)}` A) = B)
23	19, 22	syl6ibr 211	1 |- ((A e. C /\ B e. D) -> (ch -> (F` A) = B))

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221   = wceq 988   e. wcel 990  E*wmo 1414  <.cop 2456  {copab 2717  Fun wfun 3231  ` cfv 3237

This theorem is referenced by:  fvopab4g 3855  oprabval6g 4110

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-rex 1688  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-uni 2552  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fv 3253

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