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Theorem brco 4449
 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)Bx(A𝐷x x𝐶B))
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 4445 . 2 ((A V B V) → (A(𝐶𝐷)Bx(A𝐷x x𝐶B)))
41, 2, 3mp2an 402 1 (A(𝐶𝐷)Bx(A𝐷x x𝐶B))
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   ↔ wb 98  ∃wex 1378   ∈ wcel 1390  Vcvv 2551   class class class wbr 3755   ∘ 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-bnd 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:  opelco  4450  cnvco  4463  resco  4768  imaco  4769  rnco  4770  coass  4782  f1eqcocnv  5374
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