Theorem brab 4009

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 4006	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A R B <-> ch ) )
7	1, 2, 6	mp2an 402	1 |- ( A R B <-> ch )

Syntax hints:   -> wi 4   <-> wb 98   = wceq 1243   e. wcel 1393   _Vcvv 2557   class class class wbr 3764   {copab 3817

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819

This theorem is referenced by:  dftpos4  5878  enq0sym  6530  enq0ref  6531  enq0tr  6532  shftfn  9425