Theorem brabg2 33510

Description: Relation by a binary relation abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
brabg2.1	|- ( x = A -> ( ph <-> ps ) )
brabg2.2	|- ( y = B -> ( ps <-> ch ) )
brabg2.3	|- R = { <. x , y >. | ph }
brabg2.4	|- ( ch -> A e. C )

Assertion

Ref	Expression
brabg2	|- ( B e. D -> ( A R B <-> 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)   R( x, y)

Proof of Theorem brabg2

Step	Hyp	Ref	Expression
1		brabg2.3	. . . . 5 |- R = { <. x , y >. | ph }
2	1	relopabi 5245	. . . 4 |- Rel R
3	2	brrelexi 5158	. . 3 |- ( A R B -> A e. _V )
4		brabg2.1	. . . . . . 7 |- ( x = A -> ( ph <-> ps ) )
5		brabg2.2	. . . . . . 7 |- ( y = B -> ( ps <-> ch ) )
6	4, 5, 1	brabg 4994	. . . . . 6 |- ( ( A e. _V /\ B e. D ) -> ( A R B <-> ch ) )
7	6	biimpd 219	. . . . 5 |- ( ( A e. _V /\ B e. D ) -> ( A R B -> ch ) )
8	7	ex 450	. . . 4 |- ( A e. _V -> ( B e. D -> ( A R B -> ch ) ) )
9	8	com3l 89	. . 3 |- ( B e. D -> ( A R B -> ( A e. _V -> ch ) ) )
10	3, 9	mpdi 45	. 2 |- ( B e. D -> ( A R B -> ch ) )
11		brabg2.4	. . 3 |- ( ch -> A e. C )
12	4, 5, 1	brabg 4994	. . . . 5 |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ch ) )
13	12	exbiri 652	. . . 4 |- ( A e. C -> ( B e. D -> ( ch -> A R B ) ) )
14	13	com3l 89	. . 3 |- ( B e. D -> ( ch -> ( A e. C -> A R B ) ) )
15	11, 14	mpdi 45	. 2 |- ( B e. D -> ( ch -> A R B ) )
16	10, 15	impbid 202	1 |- ( B e. D -> ( A R B <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   {copab 4712

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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121

This theorem is referenced by: (None)