Theorem braba 4385

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

Hypotheses

Ref	Expression
opelopaba.1	|- A e. _V
opelopaba.2	|- B e. _V
opelopaba.3	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
braba.4	|- R = { <. x , y >. | ph }

Assertion

Ref	Expression
braba	|- ( 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)

Proof of Theorem braba

Step	Hyp	Ref	Expression
1		opelopaba.1	. 2 |- A e. _V
2		opelopaba.2	. 2 |- B e. _V
3		opelopaba.3	. . 3 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
4		braba.4	. . 3 |- R = { <. x , y >. | ph }
5	3, 4	brabga 4382	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A R B <-> ps ) )
6	1, 2, 5	mp2an 426	1 |- ( A R B <-> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   _Vcvv 2813   class class class wbr 4109   {copab 4170

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172

This theorem is referenced by: (None)