Theorem brco 4449

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 Dx /\ x CB))

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 4445	. 2 |- ((A e. _V /\ B e. _V) -> (A( C o. D)B <-> E.x(A Dx /\ x CB)))
4	1, 2, 3	mp2an 402	1 |- (A( C o. D)B <-> E.x(A Dx /\ x CB))

Syntax hints:   /\ wa 97   <-> wb 98  E.wex 1378   e. wcel 1390   _Vcvv 2551   class class class wbr 3755   o. ccom 4292

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-co 4297

This theorem is referenced by:  opelco  4450  cnvco  4463  resco  4768  imaco  4769  rnco  4770  coass  4782  f1eqcocnv  5374