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Theorem brcog 4445
Description: Ordered pair membership in a composition. (Contributed by NM, 24-Feb-2015.)
Assertion
Ref Expression
brcog  V  W  C  o.  D  D  C
Distinct variable groups:   ,   ,   , C   , D
Allowed substitution hints:    V()    W()

Proof of Theorem brcog
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 3758 . . . 4  D  D
2 breq2 3759 . . . 4  C  C
31, 2bi2anan9 538 . . 3  D  C  D  C
43exbidv 1703 . 2  D  C  D  C
5 df-co 4297 . 2  C  o.  D  { <. , 
>.  |  D  C }
64, 5brabga 3992 1  V  W  C  o.  D  D  C
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   wceq 1242  wex 1378   wcel 1390   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:  opelco2g  4446  brcogw  4447  brco  4449  brcodir  4655  foeqcnvco  5373  brtpos2  5807  ertr  6057
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