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Theorem opabresex2d 6861
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 449 . . 3 (𝜑 → (𝑥(𝑊𝐺)𝑦𝜓))
32alrimivv 2005 . 2 (𝜑 → ∀𝑥𝑦(𝑥(𝑊𝐺)𝑦𝜓))
4 opabresex2d.2 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ 𝑉)
5 opabbrex 6860 . 2 ((∀𝑥𝑦(𝑥(𝑊𝐺)𝑦𝜓) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} ∈ V)
63, 4, 5syl2anc 696 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1630  wcel 2139  Vcvv 3340   class class class wbr 4804  {copab 4864  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-v 3342  df-in 3722  df-ss 3729  df-opab 4865
This theorem is referenced by:  mptmpt2opabbrd  7416
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