Theorem elopabi 6283

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 4805	. . . 4 |- Rel { <. x , y >. | ph }
2		1st2nd 6269	. . . 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 2283	. 2 |- ( A e. { <. x , y >. | ph } -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. { <. x , y >. | ph } )
6		1stexg 6255	. . 3 |- ( A e. { <. x , y >. | ph } -> ( 1st ` A ) e. _V )
7		2ndexg 6256	. . 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 4315	. . 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 1373   e. wcel 2176   _Vcvv 2772   <.cop 3636   {copab 4105   Rel wrel 4681   ` cfv 5272   1stc1st 6226   2ndc2nd 6227

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-un 4481

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-mpt 4108  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-fo 5278  df-fv 5280  df-1st 6228  df-2nd 6229

This theorem is referenced by:  exmidapne  7374  aprcl  8721  aptap  8725