Theorem brab 4367

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 4363	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A R B <-> ch ) )
7	1, 2, 6	mp2an 426	1 |- ( A R B <-> ch )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   e. wcel 2202   _Vcvv 2802   class class class wbr 4088   {copab 4149

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151

This theorem is referenced by:  dftpos4  6428  enq0sym  7651  enq0ref  7652  enq0tr  7653  shftfn  11384