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Theorem opelco 5719
Description: Ordered pair membership in a composition. (Contributed by NM, 27-Dec-1996.) (Revised by Mario Carneiro, 24-Feb-2015.)
Hypotheses
Ref Expression
opelco.1 𝐴 ∈ V
opelco.2 𝐵 ∈ V
Assertion
Ref Expression
opelco (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷

Proof of Theorem opelco
StepHypRef Expression
1 df-br 5043 . 2 (𝐴(𝐶𝐷)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷))
2 opelco.1 . . 3 𝐴 ∈ V
3 opelco.2 . . 3 𝐵 ∈ V
42, 3brco 5718 . 2 (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
51, 4bitr3i 280 1 (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399  wex 1781  wcel 2114  Vcvv 3469  cop 4545   class class class wbr 5042  ccom 5536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-co 5541
This theorem is referenced by:  dmcoss  5820  dmcosseq  5822  cotrg  5949  coiun  6087  co02  6091  coi1  6093  coass  6096  fmptco  6873  dftpos4  7898  fmptcof2  30410  cnvco1  33069  cnvco2  33070  txpss3v  33413  dffun10  33449  xrnss3v  35743  coiun1  40284
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