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Theorem opelco2g 5874
Description: Ordered pair membership in a composition. (Contributed by NM, 27-Jan-1997.) (Revised by Mario Carneiro, 24-Feb-2015.)
Assertion
Ref Expression
opelco2g ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐷 ∧ ⟨𝑥, 𝐵⟩ ∈ 𝐶)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem opelco2g
StepHypRef Expression
1 brcog 5873 . 2 ((𝐴𝑉𝐵𝑊) → (𝐴(𝐶𝐷)𝐵 ↔ ∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵)))
2 df-br 5153 . 2 (𝐴(𝐶𝐷)𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷))
3 df-br 5153 . . . 4 (𝐴𝐷𝑥 ↔ ⟨𝐴, 𝑥⟩ ∈ 𝐷)
4 df-br 5153 . . . 4 (𝑥𝐶𝐵 ↔ ⟨𝑥, 𝐵⟩ ∈ 𝐶)
53, 4anbi12i 626 . . 3 ((𝐴𝐷𝑥𝑥𝐶𝐵) ↔ (⟨𝐴, 𝑥⟩ ∈ 𝐷 ∧ ⟨𝑥, 𝐵⟩ ∈ 𝐶))
65exbii 1842 . 2 (∃𝑥(𝐴𝐷𝑥𝑥𝐶𝐵) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐷 ∧ ⟨𝑥, 𝐵⟩ ∈ 𝐶))
71, 2, 63bitr3g 312 1 ((𝐴𝑉𝐵𝑊) → (⟨𝐴, 𝐵⟩ ∈ (𝐶𝐷) ↔ ∃𝑥(⟨𝐴, 𝑥⟩ ∈ 𝐷 ∧ ⟨𝑥, 𝐵⟩ ∈ 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wex 1773  wcel 2098  cop 4638   class class class wbr 5152  ccom 5686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-co 5691
This theorem is referenced by:  dfco2  6254  dmfco  6999  dfatdmfcoafv2  46663
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