Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  brcog Structured version   Unicode version

Theorem brcog 5031
 Description: Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
Assertion
Ref Expression
brcog
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem brcog
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4207 . . . 4
2 breq2 4208 . . . 4
31, 2bi2anan9 844 . . 3
43exbidv 1636 . 2
5 df-co 4879 . 2
64, 5brabga 4461 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wex 1550   wceq 1652   wcel 1725   class class class wbr 4204   ccom 4874 This theorem is referenced by:  opelco2g  5032  brcogw  5033  brco  5035  brcodir  5245  brtpos2  6477  ertr  6912  znleval  16827  relexpindlem  25131  opelco3  25395  funressnfv  27959 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-co 4879
 Copyright terms: Public domain W3C validator