Theorem fvopab3 4740

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 1957	. . . . 5 |- (x = A -> (x e. C <-> A e. C))
3		fvopab3.2	. . . . 5 |- (x = A -> (ph <-> ps))
4	2, 3	anbi12d 690	. . . 4 |- (x = A -> ((x e. C /\ ph) <-> (A e. C /\ ps)))
5		fvopab3.3	. . . . 5 |- (y = B -> (ps <-> ch))
6	5	anbi2d 678	. . . 4 |- (y = B -> ((A e. C /\ ps) <-> (A e. C /\ ch)))
7	4, 6	opelopabg 3567	. . 3 |- ((A e. C /\ B e. _V) -> (<.A, B>. e. {<.x, y>. | (x e. C /\ ph)} <-> (A e. C /\ ch)))
8	1, 7	mpan2 760	. 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 4548	. . . 4 |- F Fn C
12	1	fnopfvb 4713	. . . 4 |- ((F Fn C /\ A e. C) -> ((F` A) = B <-> <.A, B>. e. F))
13	11, 12	mpan 759	. . 3 |- (A e. C -> ((F` A) = B <-> <.A, B>. e. F))
14	10	eleq2i 1961	. . 3 |- (<.A, B>. e. F <-> <.A, B>. e. {<.x, y>. | (x e. C /\ ph)})
15	13, 14	syl6bb 595	. 2 |- (A e. C -> ((F` A) = B <-> <.A, B>. e. {<.x, y>. | (x e. C /\ ph)}))
16		ibar 705	. 2 |- (A e. C -> (ch <-> (A e. C /\ ch)))
17	8, 15, 16	3bitr4d 609	1 |- (A e. C -> ((F` A) = B <-> ch))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E!weu 1771  _Vcvv 2292  <.cop 3046  {copab 3395   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  recmulpq 6222

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-ral 2109  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-fn 4009  df-fv 4014

Copyright terms: Public domain