Theorem brabga 4352

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 4084	. . 3 |- ( A R B <-> <. A , B >. e. R )
2		brabga.2	. . . 4 |- R = { <. x , y >. | ph }
3	2	eleq2i 2296	. . 3 |- ( <. A , B >. e. R <-> <. A , B >. e. { <. x , y >. | ph } )
4	1, 3	bitri 184	. 2 |- ( A R B <-> <. A , B >. e. { <. x , y >. | ph } )
5		opelopabga.1	. . 3 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
6	5	opelopabga 4351	. 2 |- ( ( A e. V /\ B e. W ) -> ( <. A , B >. e. { <. x , y >. | ph } <-> ps ) )
7	4, 6	bitrid 192	1 |- ( ( A e. V /\ B e. W ) -> ( A R B <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   e. wcel 2200   <.cop 3669   class class class wbr 4083   {copab 4144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146

This theorem is referenced by:  braba  4355  brabg  4357  epelg  4381  brcog  4889  fmptco  5801  ofrfval  6227  clim  11792  isstruct2im  13042  isstruct2r  13043  eqgval  13760  dvdsrd  14058  isausgren  15965