Theorem brco 4833

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 4829	. 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 1503   e. wcel 2164   _Vcvv 2760   class class class wbr 4029   o. ccom 4663

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-co 4668

This theorem is referenced by:  opelco  4834  cnvco  4847  resco  5170  imaco  5171  rnco  5172  coass  5184  f1eqcocnv  5834