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Theorem elopabi 7735
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 5669 . . . 4 Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2 1st2nd 7713 . . . 4 ((Rel {⟨𝑥, 𝑦⟩ ∣ 𝜑} ∧ 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
31, 2mpan 689 . . 3 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
4 id 22 . . 3 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
53, 4eqeltrrd 2913 . 2 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
6 fvex 6656 . . 3 (1st𝐴) ∈ V
7 fvex 6656 . . 3 (2nd𝐴) ∈ V
8 elopabi.1 . . 3 (𝑥 = (1st𝐴) → (𝜑𝜓))
9 elopabi.2 . . 3 (𝑦 = (2nd𝐴) → (𝜓𝜒))
106, 7, 8, 9opelopab 5402 . 2 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜒)
115, 10sylib 221 1 (𝐴 ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} → 𝜒)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1538  wcel 2115  cop 4546  {copab 5101  Rel wrel 5533  cfv 6328  1st c1st 7662  2nd c2nd 7663
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6287  df-fun 6330  df-fv 6336  df-1st 7664  df-2nd 7665
This theorem is referenced by:  vciOLD  28323  sat1el2xp  32634  drngoi  35275  dicelval1sta  38369
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