Theorem brabga 4027

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 3794	. . 3 |- ( A R B <-> <. A , B >. e. R )
2		brabga.2	. . . 4 |- R = { <. x , y >. | ph }
3	2	eleq2i 2146	. . 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 4026	. 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 1285   e. wcel 1434   <.cop 3409   class class class wbr 3793   {copab 3846

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848

This theorem is referenced by:  braba  4030  brabg  4032  epelg  4053  brcog  4530  fmptco  5362  ofrfval  5751  clim  10258