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Theorem brco 5827
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 5823 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
41, 2, 3mp2an 691 1 (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  wex 1782  wcel 2107  Vcvv 3444   class class class wbr 5106  ccom 5638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-co 5643
This theorem is referenced by:  opelco  5828  cnvco  5842  cotrg  6062  cotrgOLD  6063  resco  6203  imaco  6204  rnco  6205  coass  6218  dfpo2  6249  dffv2  6937  foeqcnvco  7247  f1eqcocnv  7248  f1eqcocnvOLD  7249  ttrclss  9661  rtrclreclem3  14951  imasleval  17428  ustuqtop4  23612  metustexhalf  23928  dftr6  34380  coep  34381  coepr  34382  brtxp  34511  pprodss4v  34515  brpprod  34516  sscoid  34544  elfuns  34546  brimg  34568  brapply  34569  brcup  34570  brcap  34571  brsuccf  34572  funpartlem  34573  brrestrict  34580  dfrecs2  34581  dfrdg4  34582  cnvssco  41966
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