Theorem fvopab3 3777

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

Hypotheses

Ref	Expression
fvopab3.1	|- B e. V
fvopab3.2	|- (x = A -> (ph <-> ps))
fvopab3.3	|- (y = B -> (ps <-> ch))
fvopab3.4	|- (x e. C -> E!yph)
fvopab3.5	|- F = {<.x, y>. | (x e. C /\ ph)}

Assertion

Ref	Expression
fvopab3	|- (A e. C -> ((F` A) = B <-> ch))

Distinct variable groups:   x,y,A   x,B,y   x,C,y   ch,x,y

Proof of Theorem fvopab3

Step	Hyp	Ref	Expression
1		fvopab3.1	. . 3 |- B e. V
2		eleq1 1534	. . . . 5 |- (x = A -> (x e. C <-> A e. C))
3		fvopab3.2	. . . . 5 |- (x = A -> (ph <-> ps))
4	2, 3	anbi12d 628	. . . 4 |- (x = A -> ((x e. C /\ ph) <-> (A e. C /\ ps)))
5		fvopab3.3	. . . . 5 |- (y = B -> (ps <-> ch))
6	5	anbi2d 616	. . . 4 |- (y = B -> ((A e. C /\ ps) <-> (A e. C /\ ch)))
7	4, 6	opelopabg 2817	. . 3 |- ((A e. C /\ B e. V) -> (<.A, B>. e. {<.x, y>. | (x e. C /\ ph)} <-> (A e. C /\ ch)))
8	1, 7	mpan2 696	. 2 |- (A e. C -> (<.A, B>. e. {<.x, y>. | (x e. C /\ ph)} <-> (A e. C /\ ch)))
9		fvopab3.4	. . . . 5 |- (x e. C -> E!yph)
10		fvopab3.5	. . . . 5 |- F = {<.x, y>. | (x e. C /\ ph)}
11	9, 10	fnopab 3617	. . . 4 |- F Fn C
12	1	fnopfvb 3754	. . . 4 |- ((F Fn C /\ A e. C) -> ((F` A) = B <-> <.A, B>. e. F))
13	11, 12	mpan 695	. . 3 |- (A e. C -> ((F` A) = B <-> <.A, B>. e. F))
14	10	eleq2i 1538	. . 3 |- (<.A, B>. e. F <-> <.A, B>. e. {<.x, y>. | (x e. C /\ ph)})
15	13, 14	syl6bb 536	. 2 |- (A e. C -> ((F` A) = B <-> <.A, B>. e. {<.x, y>. | (x e. C /\ ph)}))
16		ibar 643	. 2 |- (A e. C -> (ch <-> (A e. C /\ ch)))
17	8, 15, 16	3bitr4d 550	1 |- (A e. C -> ((F` A) = B <-> ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E!weu 1380  Vcvv 1811  <.cop 2411  {copab 2666   Fn wfn 3177  ` cfv 3182

This theorem is referenced by:  recmulpq 5070

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-ral 1649  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-fn 3193  df-fv 3198

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