Theorem opelopab3 33511

Description: Ordered pair membership in an ordered pair class abstraction, with a reduced hypothesis. (Contributed by Jeff Madsen, 29-May-2011.)

Hypotheses

Ref	Expression
opelopab3.1	|- ( x = A -> ( ph <-> ps ) )
opelopab3.2	|- ( y = B -> ( ps <-> ch ) )
opelopab3.3	|- ( ch -> A e. C )

Assertion

Ref	Expression
opelopab3	|- ( B e. D -> ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) )

Distinct variable groups:   x, A, y   x, B, y   ch, x, y

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

Proof of Theorem opelopab3

Step	Hyp	Ref	Expression
1		elopaelxp 5191	. . . . 5 |- ( <. A , B >. e. { <. x , y >. | ph } -> <. A , B >. e. ( _V X. _V ) )
2		opelxp1 5150	. . . . 5 |- ( <. A , B >. e. ( _V X. _V ) -> A e. _V )
3	1, 2	syl 17	. . . 4 |- ( <. A , B >. e. { <. x , y >. | ph } -> A e. _V )
4	3	anim1i 592	. . 3 |- ( ( <. A , B >. e. { <. x , y >. | ph } /\ B e. D ) -> ( A e. _V /\ B e. D ) )
5	4	ancoms 469	. 2 |- ( ( B e. D /\ <. A , B >. e. { <. x , y >. | ph } ) -> ( A e. _V /\ B e. D ) )
6		opelopab3.3	. . . . 5 |- ( ch -> A e. C )
7		elex 3212	. . . . 5 |- ( A e. C -> A e. _V )
8	6, 7	syl 17	. . . 4 |- ( ch -> A e. _V )
9	8	anim1i 592	. . 3 |- ( ( ch /\ B e. D ) -> ( A e. _V /\ B e. D ) )
10	9	ancoms 469	. 2 |- ( ( B e. D /\ ch ) -> ( A e. _V /\ B e. D ) )
11		opelopab3.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
12		opelopab3.2	. . 3 |- ( y = B -> ( ps <-> ch ) )
13	11, 12	opelopabg 4993	. 2 |- ( ( A e. _V /\ B e. D ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) )
14	5, 10, 13	pm5.21nd 941	1 |- ( B e. D -> ( <. A , B >. e. { <. x , y >. | ph } <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   <.cop 4183   {copab 4712   X. cxp 5112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-opab 4713  df-xp 5120

This theorem is referenced by: (None)