Theorem braba 4314

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 4311	. 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 1373   e. wcel 2176   _Vcvv 2772   class class class wbr 4045   {copab 4105

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107

This theorem is referenced by: (None)