Theorem brco 4893

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 4889	. 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 426	1 |- ( A ( C o. D ) B <-> E. x ( A D x /\ x C B ) )

Syntax hints:   /\ wa 104   <-> wb 105   E.wex 1538   e. wcel 2200   _Vcvv 2799   class class class wbr 4083   o. ccom 4723

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-co 4728

This theorem is referenced by:  opelco  4894  cnvco  4907  resco  5233  imaco  5234  rnco  5235  coass  5247  f1eqcocnv  5915