Theorem elopabi 4101

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 3256	. . 3 |- Rel {<.x, y>. | ph}
2		1st2nd 4092	. . 3 |- ((Rel {<.x, y>. | ph} /\ A e. {<.x, y>. | ph}) -> A = <.(1st` A), (2nd` A)>.)
3	1, 2	mpan 693	. 2 |- (A e. {<.x, y>. | ph} -> A = <.(1st` A), (2nd` A)>.)
4		eleq1 1526	. . . 4 |- (A = <.(1st` A), (2nd` A)>. -> (A e. {<.x, y>. | ph} <-> <.(1st` A), (2nd` A)>. e. {<.x, y>. | ph}))
5		fvex 3717	. . . . 5 |- (1st` A) e. V
6		fvex 3717	. . . . 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 2809	. . . 4 |- (<.(1st` A), (2nd` A)>. e. {<.x, y>. | ph} <-> ch)
10	4, 9	syl6bb 534	. . 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 953   e. wcel 955  <.cop 2401  {copab 2656  Rel wrel 3165  ` cfv 3172  1stc1st 4061  2ndc2nd 4062

This theorem is referenced by:  eloprabi 4102  msflem 7742  bcthlem15 7947  ringi 8079  vci 8104  hgralem 10606

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fv 3188  df-1st 4063  df-2nd 4064

Copyright terms: Public domain