Theorem elopabi 5059

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 4104	. . 3 |- Rel {<.x, y>. | ph}
2		1st2nd 5048	. . 3 |- ((Rel {<.x, y>. | ph} /\ A e. {<.x, y>. | ph}) -> A = <.(1st` A), (2nd` A)>.)
3	1, 2	mpan 759	. 2 |- (A e. {<.x, y>. | ph} -> A = <.(1st` A), (2nd` A)>.)
4		eleq1 1957	. . . 4 |- (A = <.(1st` A), (2nd` A)>. -> (A e. {<.x, y>. | ph} <-> <.(1st` A), (2nd` A)>. e. {<.x, y>. | ph}))
5		fvex 4689	. . . . 5 |- (1st` A) e. _V
6		fvex 4689	. . . . 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 3570	. . . 4 |- (<.(1st` A), (2nd` A)>. e. {<.x, y>. | ph} <-> ch)
10	4, 9	syl6bb 595	. . 3 |- (A = <.(1st` A), (2nd` A)>. -> (A e. {<.x, y>. | ph} <-> ch))
11	10	biimpcd 172	. 2 |- (A e. {<.x, y>. | ph} -> (A = <.(1st` A), (2nd` A)>. -> ch))
12	3, 11	mpd 29	1 |- (A e. {<.x, y>. | ph} -> ch)

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  <.cop 3046  {copab 3395  Rel wrel 3991  ` cfv 3998  1stc1st 5018  2ndc2nd 5019

This theorem is referenced by:  eloprabi 5060  msflem 9080  bcthlem15 9291  ringi 9466  drngi 9493  vci 9499  com2i 14765  vri 14834  topgrpi 14972  heiborlem30 15984  hgralem 16292

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-13 1311  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  ax-un 3790

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-fv 4014  df-1st 5020  df-2nd 5021

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