Theorem elopabi 4123

Description: A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors.

Hypotheses

Ref	Expression
elopabi.1	|- (x = (1st` A) -> (ph <-> ps))
elopabi.2	|- (y = (2nd` A) -> (ps <-> ch))

Assertion

Ref	Expression
elopabi	|- (A e. {<.x, y>. | ph} -> ch)

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

Proof of Theorem elopabi

Step	Hyp	Ref	Expression
1		relopab 3272	. . 3 |- Rel {<.x, y>. | ph}
2		1st2nd 4114	. . 3 |- ((Rel {<.x, y>. | ph} /\ A e. {<.x, y>. | ph}) -> A = <.(1st` A), (2nd` A)>.)
3	1, 2	mpan 697	. 2 |- (A e. {<.x, y>. | ph} -> A = <.(1st` A), (2nd` A)>.)
4		eleq1 1537	. . . 4 |- (A = <.(1st` A), (2nd` A)>. -> (A e. {<.x, y>. | ph} <-> <.(1st` A), (2nd` A)>. e. {<.x, y>. | ph}))
5		fvex 3738	. . . . 5 |- (1st` A) e. V
6		fvex 3738	. . . . 5 |- (2nd` A) e. V
7		elopabi.1	. . . . 5 |- (x = (1st` A) -> (ph <-> ps))
8		elopabi.2	. . . . 5 |- (y = (2nd` A) -> (ps <-> ch))
9	5, 6, 7, 8	opelopab 2826	. . . 4 |- (<.(1st` A), (2nd` A)>. e. {<.x, y>. | ph} <-> ch)
10	4, 9	syl6bb 538	. . 3 |- (A = <.(1st` A), (2nd` A)>. -> (A e. {<.x, y>. | ph} <-> ch))
11	10	biimpcd 155	. 2 |- (A e. {<.x, y>. | ph} -> (A = <.(1st` A), (2nd` A)>. -> ch))
12	3, 11	mpd 26	1 |- (A e. {<.x, y>. | ph} -> ch)

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  <.cop 2415  {copab 2671  Rel wrel 3181  ` cfv 3188  1stc1st 4083  2ndc2nd 4084

This theorem is referenced by:  eloprabi 4124  msflem 7800  bcthlem15 8010  ringi 8138  vci 8163  hgralem 10741

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-1st 4085  df-2nd 4086

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