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Theorem elopabi 8072
Description: A consequence of membership in an ordered-pair class abstraction, using ordered pair extractors. (Contributed by NM, 29-Aug-2006.)
Hypotheses
Ref Expression
elopabi.1 (𝑥 = (1st𝐴) → (𝜑𝜓))
elopabi.2 (𝑦 = (2nd𝐴) → (𝜓𝜒))
Assertion
Ref Expression
elopabi (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)
Distinct variable groups:   𝑥,𝑦,𝐴   𝜒,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem elopabi
StepHypRef Expression
1 relopabv 5827 . . . 4 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 1st2nd 8049 . . . 4 ((Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
31, 2mpan 688 . . 3 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
4 id 22 . . 3 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
53, 4eqeltrrd 2830 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
6 fvex 6915 . . 3 (1st𝐴) ∈ V
7 fvex 6915 . . 3 (2nd𝐴) ∈ V
8 elopabi.1 . . 3 (𝑥 = (1st𝐴) → (𝜑𝜓))
9 elopabi.2 . . 3 (𝑦 = (2nd𝐴) → (𝜓𝜒))
106, 7, 8, 9opelopab 5548 . 2 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)
115, 10sylib 217 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1533  wcel 2098  cop 4638  {copab 5214  Rel wrel 5687  cfv 6553  1st c1st 7997  2nd c2nd 7998
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-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
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-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  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-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-iota 6505  df-fun 6555  df-fv 6561  df-1st 7999  df-2nd 8000
This theorem is referenced by:  vciOLD  30391  sat1el2xp  35022  drngoi  37457  dicelval1sta  40692
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