Theorem fvopab3ig 4741

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 1957	. . . . . . . . 9 |- (x = A -> (x e. C <-> A e. C))
2		fvopab3ig.1	. . . . . . . . 9 |- (x = A -> (ph <-> ps))
3	1, 2	anbi12d 690	. . . . . . . 8 |- (x = A -> ((x e. C /\ ph) <-> (A e. C /\ ps)))
4		fvopab3ig.2	. . . . . . . . 9 |- (y = B -> (ps <-> ch))
5	4	anbi2d 678	. . . . . . . 8 |- (y = B -> ((A e. C /\ ps) <-> (A e. C /\ ch)))
6	3, 5	opelopabg 3567	. . . . . . 7 |- ((A e. C /\ B e. D) -> (<.A, B>. e. {<.x, y>. | (x e. C /\ ph)} <-> (A e. C /\ ch)))
7	6	biimpar 461	. . . . . 6 |- (((A e. C /\ B e. D) /\ (A e. C /\ ch)) -> <.A, B>. e. {<.x, y>. | (x e. C /\ ph)})
8	7	exp43 415	. . . . 5 |- (A e. C -> (B e. D -> (A e. C -> (ch -> <.A, B>. e. {<.x, y>. | (x e. C /\ ph)}))))
9	8	pm2.43a 80	. . . 4 |- (A e. C -> (B e. D -> (ch -> <.A, B>. e. {<.x, y>. | (x e. C /\ ph)})))
10	9	imp 377	. . 3 |- ((A e. C /\ B e. D) -> (ch -> <.A, B>. e. {<.x, y>. | (x e. C /\ ph)}))
11		funopab 4455	. . . . . 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 1829	. . . . . . 7 |- (E*y(x e. C /\ ph) <-> (x e. C -> E*yph))
14	12, 13	mpbir 207	. . . . . 6 |- E*y(x e. C /\ ph)
15	11, 14	mpgbir 1334	. . . . 5 |- Fun {<.x, y>. | (x e. C /\ ph)}
16		funopfvg 4711	. . . . 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 760	. . . 4 |- (B e. D -> (<.A, B>. e. {<.x, y>. | (x e. C /\ ph)} -> ({<.x, y>. | (x e. C /\ ph)}` A) = B))
18	17	adantl 424	. . 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 30	. 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 4682	. . 3 |- (F` A) = ({<.x, y>. | (x e. C /\ ph)}` A)
22	21	eqeq1i 1891	. 2 |- ((F` A) = B <-> ({<.x, y>. | (x e. C /\ ph)}` A) = B)
23	19, 22	syl6ibr 230	1 |- ((A e. C /\ B e. D) -> (ch -> (F` A) = B))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E*wmo 1772  <.cop 3046  {copab 3395  Fun wfun 3992  ` cfv 3998

This theorem is referenced by:  fvopab4g 4742  oprabval6g 4962

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fv 4014

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