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Theorem brco 4680
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 ∈ V
opelco.2 𝐵 ∈ V
Assertion
Ref Expression
brco (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷

Proof of Theorem brco
StepHypRef Expression
1 opelco.1 . 2 𝐴 ∈ V
2 opelco.2 . 2 𝐵 ∈ V
3 brcog 4676 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
41, 2, 3mp2an 422 1 (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  wex 1453  wcel 1465  Vcvv 2660   class class class wbr 3899  ccom 4513
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-co 4518
This theorem is referenced by:  opelco  4681  cnvco  4694  resco  5013  imaco  5014  rnco  5015  coass  5027  f1eqcocnv  5660
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