Theorem elopabi 6873

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 4965	. . . 4 |- Rel { <. x , y >. | ph }
2		1st2nd 6858	. . . 4 |- ( ( Rel { <. x , y >. | ph } /\ A e. { <. x , y >. | ph } ) -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
3	1, 2	mpan 684	. . 3 |- ( A e. { <. x , y >. | ph } -> A = <. ( 1st ` A ) , ( 2nd ` A ) >. )
4		id 22	. . 3 |- ( A e. { <. x , y >. | ph } -> A e. { <. x , y >. | ph } )
5	3, 4	eqeltrrd 2550	. 2 |- ( A e. { <. x , y >. | ph } -> <. ( 1st ` A ) , ( 2nd ` A ) >. e. { <. x , y >. | ph } )
6		fvex 5889	. . 3 |- ( 1st ` A ) e. _V
7		fvex 5889	. . 3 |- ( 2nd ` A ) e. _V
8		elopabi.1	. . 3 |- ( x = ( 1st ` A ) -> ( ph <-> ps ) )
9		elopabi.2	. . 3 |- ( y = ( 2nd ` A ) -> ( ps <-> ch ) )
10	6, 7, 8, 9	opelopab 4723	. 2 |- ( <. ( 1st ` A ) , ( 2nd ` A ) >. e. { <. x , y >. | ph } <-> ch )
11	5, 10	sylib 201	1 |- ( A e. { <. x , y >. | ph } -> ch )

Syntax hints:   -> wi 4   <-> wb 189   = wceq 1452   e. wcel 1904   <.cop 3965   {copab 4453   Rel wrel 4844   ` cfv 5589   1stc1st 6810   2ndc2nd 6811

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-1st 6812  df-2nd 6813

This theorem is referenced by:  uhgrac  25111  wlkelwrd  25337  drngoi  26216  vci  26248  dicelval1sta  34826