Theorem brab 4415

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 4411	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A R B <-> ch ) )
7	1, 2, 6	mp2an 430	1 |- ( A R B <-> ch )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   class class class wbr 4130   {copab 4191

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193

This theorem is used by:  dftpos4  6534  enq0sym  7799  enq0ref  7800  enq0tr  7801  shftfn  11589