Theorem brabga 4082

Description: The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
opelopabga.1	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
brabga.2	|- R = { <. x , y >. | ph }

Assertion

Ref	Expression
brabga	|- ( ( A e. V /\ B e. W ) -> ( A R B <-> ps ) )

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

Allowed substitution hints:   ph( x, y)   R( x, y)   V( x, y)   W( x, y)

Proof of Theorem brabga

Step	Hyp	Ref	Expression
1		df-br 3838	. . 3 |- ( A R B <-> <. A , B >. e. R )
2		brabga.2	. . . 4 |- R = { <. x , y >. | ph }
3	2	eleq2i 2154	. . 3 |- ( <. A , B >. e. R <-> <. A , B >. e. { <. x , y >. | ph } )
4	1, 3	bitri 182	. 2 |- ( A R B <-> <. A , B >. e. { <. x , y >. | ph } )
5		opelopabga.1	. . 3 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
6	5	opelopabga 4081	. 2 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ps ) )
7	4, 6	syl5bb 190	1 |- ( ( A e. V /\ B e. W ) -> ( A R B <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1289   e. wcel 1438   <.cop 3444   class class class wbr 3837   {copab 3890

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892

This theorem is referenced by:  braba  4085  brabg  4087  epelg  4108  brcog  4591  fmptco  5448  ofrfval  5846  clim  10633