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Theorem opelco 5884
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 5148 . 2 (𝐴(𝐶𝐷)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷))
2 opelco.1 . . 3 𝐴 ∈ V
3 opelco.2 . . 3 𝐵 ∈ V
42, 3brco 5883 . 2 (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
51, 4bitr3i 277 1 (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  wex 1775  wcel 2105  Vcvv 3477  cop 4636   class class class wbr 5147  ccom 5692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-co 5697
This theorem is referenced by:  dmcoss  5987  dmcosseq  5989  dmcosseqOLD  5990  cotrgOLDOLD  6131  coiun  6277  co02  6281  coi1  6283  coass  6286  fmptco  7148  dftpos4  8268  ttrcltr  9753  fmptcof2  32673  cnvco1  35738  cnvco2  35739  txpss3v  35859  dffun10  35895  xrnss3v  38353  coiun1  43641
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