Theorem brab 4257

Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.)

Hypotheses

Ref	Expression
opelopab.1	|- A e. _V
opelopab.2	|- B e. _V
opelopab.3	|- ( x = A -> ( ph <-> ps ) )
opelopab.4	|- ( y = B -> ( ps <-> ch ) )
brab.5	|- R = { <. x , y >. | ph }

Assertion

Ref	Expression
brab	|- ( A R B <-> ch )

Distinct variable groups:   x, y, A   x, B, y   ch, x, y

Allowed substitution hints:   ph( x, y)   ps( x, y)   R( x, y)

Proof of Theorem brab

Step	Hyp	Ref	Expression
1		opelopab.1	. 2 |- A e. _V
2		opelopab.2	. 2 |- B e. _V
3		opelopab.3	. . 3 |- ( x = A -> ( ph <-> ps ) )
4		opelopab.4	. . 3 |- ( y = B -> ( ps <-> ch ) )
5		brab.5	. . 3 |- R = { <. x , y >. | ph }
6	3, 4, 5	brabg 4254	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A R B <-> ch ) )
7	1, 2, 6	mp2an 424	1 |- ( A R B <-> ch )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   e. wcel 2141   _Vcvv 2730   class class class wbr 3989   {copab 4049

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051

This theorem is referenced by:  dftpos4  6242  enq0sym  7394  enq0ref  7395  enq0tr  7396  shftfn  10788