Theorem braba 4189

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 4186	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A R B <-> ps ) )
6	1, 2, 5	mp2an 422	1 |- ( A R B <-> ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1331   e. wcel 1480   _Vcvv 2686   class class class wbr 3929   {copab 3988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990

This theorem is referenced by: (None)