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Theorem brco 5734
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 5730 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
41, 2, 3mp2an 688 1 (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wex 1771  wcel 2105  Vcvv 3492   class class class wbr 5057  ccom 5552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-co 5557
This theorem is referenced by:  opelco  5735  cnvco  5749  resco  6096  imaco  6097  rnco  6098  coass  6111  dffv2  6749  foeqcnvco  7047  f1eqcocnv  7048  rtrclreclem3  14407  imasleval  16802  ustuqtop4  22780  metustexhalf  23093  dftr6  32883  coep  32884  coepr  32885  dfpo2  32888  brtxp  33238  pprodss4v  33242  brpprod  33243  sscoid  33271  elfuns  33273  brimg  33295  brapply  33296  brcup  33297  brcap  33298  brsuccf  33299  funpartlem  33300  brrestrict  33307  dfrecs2  33308  dfrdg4  33309  cnvssco  39844
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