Theorem elopabi 6155

Description: A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)

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

Allowed substitution hints:   ph( x, y)   ps( x, y)

Proof of Theorem elopabi

Step	Hyp	Ref	Expression
1		relopab 4725	. . . 4 |- Rel { <. x , y >. | ph }
2		1st2nd 6141	. . . 4 |- ( ( Rel { <. x , y >. | ph } /\ A e. { <. x , y >. | ph } ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
3	1, 2	mpan 421	. . 3 |- ( A e. { <. x , y >. | ph } -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
4		id 19	. . 3 |- ( A e. { <. x , y >. | ph } -> A e. { <. x , y >. | ph } )
5	3, 4	eqeltrrd 2242	. 2 |- ( A e. { <. x , y >. | ph } -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. { <. x , y >. | ph } )
6		1stexg 6127	. . 3 |- ( A e. { <. x , y >. | ph } -> ( 1st ` A ) e. _V )
7		2ndexg 6128	. . 3 |- ( A e. { <. x , y >. | ph } -> ( 2nd ` A ) e. _V )
8		elopabi.1	. . . 4 |- ( x = ( 1st ` A ) -> ( ph <-> ps ) )
9		elopabi.2	. . . 4 |- ( y = ( 2nd ` A ) -> ( ps <-> ch ) )
10	8, 9	opelopabg 4240	. . 3 |- ( ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. { <. x , y >. | ph } <-> ch ) )
11	6, 7, 10	syl2anc 409	. 2 |- ( A e. { <. x , y >. | ph } -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. { <. x , y >. | ph } <-> ch ) )
12	5, 11	mpbid 146	1 |- ( A e. { <. x , y >. | ph } -> ch )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1342   e. wcel 2135   _Vcvv 2721   <.cop 3573   {copab 4036   Rel wrel 4603   ` cfv 5182   1stc1st 6098   2ndc2nd 6099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-sbc 2947  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-fo 5188  df-fv 5190  df-1st 6100  df-2nd 6101

This theorem is referenced by:  aprcl  8535