Theorem opabresex2d 7307

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	⊢ ((𝜑 ∧ 𝑥(𝑊‘𝐺)𝑦) → 𝜓)
opabresex2d.2	⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ 𝑉)

Assertion

Ref	Expression
opabresex2d	⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ 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

Step	Hyp	Ref	Expression
1		opabresex2d.1	. . . 4 ⊢ ((𝜑 ∧ 𝑥(𝑊‘𝐺)𝑦) → 𝜓)
2	1	ex 412	. . 3 ⊢ (𝜑 → (𝑥(𝑊‘𝐺)𝑦 → 𝜓))
3	2	alrimivv 1932	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥(𝑊‘𝐺)𝑦 → 𝜓))
4		opabresex2d.2	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ 𝑉)
5		opabbrex 7306	. 2 ⊢ ((∀𝑥∀𝑦(𝑥(𝑊‘𝐺)𝑦 → 𝜓) ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ 𝑉) → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V)
6	3, 4, 5	syl2anc 583	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   ∈ wcel 2108  Vcvv 3422   class class class wbr 5070  {copab 5132  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-in 3890  df-ss 3900  df-opab 5133

This theorem is referenced by:  mptmpoopabbrd  7894