Theorem brcog 5174

Description: Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)

Assertion

Ref	Expression
brcog

Distinct variable groups:   ,   ,   ,   ,

Proof of Theorem brcog

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4455	. . . 4
2		breq2 4456	. . . 4
3	1, 2	bi2anan9 873	. . 3
4	3	exbidv 1714	. 2
5		df-co 5013	. 2
6	4, 5	brabga 4766	1

Syntax hints:  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818   class class class wbr 4452  o.ccom 5008

This theorem is referenced by:  opelco2g  5175  brcogw  5176  brco  5178  brcodir  5391  brtpos2  6980  ertr  7345  znleval  18593  fcoinvbr  27461  relexpindlem  29062  opelco3  29208  funressnfv  32213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-co 5013