Theorem brco 4718

Description: Binary relation on a composition. (Contributed by NM, 21-Sep-2004.) (Revised by Mario Carneiro, 24-Feb-2015.)

Hypotheses

Ref	Expression
opelco.1	|- A e. _V
opelco.2	|- B e. _V

Assertion

Ref	Expression
brco	|- ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) )

Distinct variable groups:   x, A   x, B   x, C   x, D

Proof of Theorem brco

Step	Hyp	Ref	Expression
1		opelco.1	. 2 |- A e. _V
2		opelco.2	. 2 |- B e. _V
3		brcog 4714	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) ) )
4	1, 2, 3	mp2an 423	1 |- ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) )

Syntax hints:   /\ wa 103   <-> wb 104   E.wex 1469   e. wcel 1481   _Vcvv 2689   class class class wbr 3937   o. ccom 4551

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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-br 3938  df-opab 3998  df-co 4556

This theorem is referenced by:  opelco  4719  cnvco  4732  resco  5051  imaco  5052  rnco  5053  coass  5065  f1eqcocnv  5700