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Theorem elopabi 7391
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 relopab 5395 . . . 4 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 1st2nd 7373 . . . 4 ((Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
31, 2mpan 708 . . 3 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
4 id 22 . . 3 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
53, 4eqeltrrd 2832 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
6 fvex 6354 . . 3 (1st𝐴) ∈ V
7 fvex 6354 . . 3 (2nd𝐴) ∈ V
8 elopabi.1 . . 3 (𝑥 = (1st𝐴) → (𝜑𝜓))
9 elopabi.2 . . 3 (𝑦 = (2nd𝐴) → (𝜓𝜒))
106, 7, 8, 9opelopab 5139 . 2 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)
115, 10sylib 208 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1624  wcel 2131  cop 4319  {copab 4856  Rel wrel 5263  cfv 6041  1st c1st 7323  2nd c2nd 7324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-iota 6004  df-fun 6043  df-fv 6049  df-1st 7325  df-2nd 7326
This theorem is referenced by:  vciOLD  27717  drngoi  34055  dicelval1sta  36970
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