Theorem brab 4307

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 4303	. 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 1364   e. wcel 2167   _Vcvv 2763   class class class wbr 4033   {copab 4093

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-br 4034  df-opab 4095

This theorem is referenced by:  dftpos4  6321  enq0sym  7499  enq0ref  7500  enq0tr  7501  shftfn  10989