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Theorem opabresex2d 6961
 Description: Restrictions of a collection of ordered pairs of related elements are sets. (Contributed by Alexander van der Vekens, 1-Nov-2017.) (Revised by AV, 15-Jan-2021.)
Hypotheses
Ref Expression
opabresex2d.1 ((𝜑𝑥(𝑊𝐺)𝑦) → 𝜓)
opabresex2d.2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ 𝑉)
Assertion
Ref Expression
opabresex2d (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} ∈ V)
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜃(𝑥,𝑦)   𝐺(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem opabresex2d
StepHypRef Expression
1 opabresex2d.1 . . . 4 ((𝜑𝑥(𝑊𝐺)𝑦) → 𝜓)
21ex 403 . . 3 (𝜑 → (𝑥(𝑊𝐺)𝑦𝜓))
32alrimivv 2027 . 2 (𝜑 → ∀𝑥𝑦(𝑥(𝑊𝐺)𝑦𝜓))
4 opabresex2d.2 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ 𝑉)
5 opabbrex 6960 . 2 ((∀𝑥𝑦(𝑥(𝑊𝐺)𝑦𝜓) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} ∈ V)
63, 4, 5syl2anc 579 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386  ∀wal 1654   ∈ wcel 2164  Vcvv 3414   class class class wbr 4875  {copab 4937  ‘cfv 6127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-v 3416  df-in 3805  df-ss 3812  df-opab 4938 This theorem is referenced by:  mptmpt2opabbrd  7516
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