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Theorem brcog 4502
 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 3767 . . . 4
2 breq2 3768 . . . 4
31, 2bi2anan9 538 . . 3
43exbidv 1706 . 2
5 df-co 4354 . 2
64, 5brabga 4001 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wb 98   wceq 1243  wex 1381   wcel 1393   class class class wbr 3764   ccom 4349 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-co 4354 This theorem is referenced by:  opelco2g  4503  brcogw  4504  brco  4506  brcodir  4712  foeqcnvco  5430  brtpos2  5866  ertr  6121
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