Theorem opabresex2d 6696

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 450	. . 3 ⊢ (𝜑 → (𝑥(𝑊‘𝐺)𝑦 → 𝜓))
3	2	alrimivv 1856	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦(𝑥(𝑊‘𝐺)𝑦 → 𝜓))
4		opabresex2d.2	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ 𝑉)
5		opabbrex 6695	. 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 693	1 ⊢ (𝜑 → {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V)

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481   ∈ wcel 1990  Vcvv 3200   class class class wbr 4653  {copab 4712  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-in 3581  df-ss 3588  df-opab 4713

This theorem is referenced by:  mptmpt2opabbrd  7248