Theorem elopabi 6359

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 4856	. . . 4 |- Rel { <. x , y >. | ph }
2		1st2nd 6343	. . . 4 |- ( ( Rel { <. x , y >. | ph } /\ A e. { <. x , y >. | ph } ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
3	1, 2	mpan 424	. . 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 2309	. 2 |- ( A e. { <. x , y >. | ph } -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. { <. x , y >. | ph } )
6		1stexg 6329	. . 3 |- ( A e. { <. x , y >. | ph } -> ( 1st ` A ) e. _V )
7		2ndexg 6330	. . 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 4362	. . 3 |- ( ( ( 1st ` A ) e. _V /\ ( 2nd ` A ) e. _V ) -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. { <. x , y >. | ph } <-> ch ) )
11	6, 7, 10	syl2anc 411	. 2 |- ( A e. { <. x , y >. | ph } -> ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. { <. x , y >. | ph } <-> ch ) )
12	5, 11	mpbid 147	1 |- ( A e. { <. x , y >. | ph } -> ch )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   e. wcel 2202   _Vcvv 2802   <.cop 3672   {copab 4149   Rel wrel 4730   ` cfv 5326   1stc1st 6300   2ndc2nd 6301

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-1st 6302  df-2nd 6303

This theorem is referenced by:  exmidapne  7478  aprcl  8825  aptap  8829