Theorem elopabi 6391

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 4881	. . . 4 |- Rel { <. x , y >. | ph }
2		1st2nd 6375	. . . 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 2310	. 2 |- ( A e. { <. x , y >. | ph } -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. { <. x , y >. | ph } )
6		1stexg 6361	. . 3 |- ( A e. { <. x , y >. | ph } -> ( 1st ` A ) e. _V )
7		2ndexg 6362	. . 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 4386	. . 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 1398   e. wcel 2203   _Vcvv 2813   <.cop 3692   {copab 4170   Rel wrel 4754   ` cfv 5352   1stc1st 6332   2ndc2nd 6333

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fo 5358  df-fv 5360  df-1st 6334  df-2nd 6335

This theorem is referenced by:  exmidapne  7574  aprcl  8920  aptap  8924