Theorem brabg2 30181

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 5118	. . . 4 |- Rel R
3	2	brrelexi 5030	. . 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 4756	. . . . . 6 |- ( ( A e. _V /\ B e. D ) -> ( A R B <-> ch ) )
7	6	biimpd 207	. . . . 5 |- ( ( A e. _V /\ B e. D ) -> ( A R B -> ch ) )
8	7	ex 434	. . . 4 |- ( A e. _V -> ( B e. D -> ( A R B -> ch ) ) )
9	8	com3l 81	. . 3 |- ( B e. D -> ( A R B -> ( A e. _V -> ch ) ) )
10	3, 9	mpdi 42	. 2 |- ( B e. D -> ( A R B -> ch ) )
11		brabg2.4	. . 3 |- ( ch -> A e. C )
12	4, 5, 1	brabg 4756	. . . . 5 |- ( ( A e. C /\ B e. D ) -> ( A R B <-> ch ) )
13	12	exbiri 622	. . . 4 |- ( A e. C -> ( B e. D -> ( ch -> A R B ) ) )
14	13	com3l 81	. . 3 |- ( B e. D -> ( ch -> ( A e. C -> A R B ) ) )
15	11, 14	mpdi 42	. 2 |- ( B e. D -> ( ch -> A R B ) )
16	10, 15	impbid 191	1 |- ( B e. D -> ( A R B <-> ch ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1383   e. wcel 1804   _Vcvv 3095   class class class wbr 4437   {copab 4494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-br 4438  df-opab 4496  df-xp 4995  df-rel 4996

This theorem is referenced by: (None)