Theorem opabresex2 7327

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.) Add disjoint variable conditions betweem 𝑊, 𝐺 and 𝑥, 𝑦 to remove hypotheses. (Revised by SN, 13-Dec-2024.)

Assertion

Ref	Expression
opabresex2	⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V

Distinct variable groups:   𝑥,𝑊   𝑦,𝑊   𝑥,𝐺   𝑦,𝐺

Allowed substitution hints:   𝜃(𝑥,𝑦)

Proof of Theorem opabresex2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fvex 6787	. 2 ⊢ (𝑊‘𝐺) ∈ V
2		elopabran 5475	. . 3 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} → 𝑧 ∈ (𝑊‘𝐺))
3	2	ssriv 3925	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ⊆ (𝑊‘𝐺)
4	1, 3	ssexi 5246	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V

Syntax hints:   ∧ wa 396   ∈ wcel 2106  Vcvv 3432   class class class wbr 5074  {copab 5136  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-sn 4562  df-pr 4564  df-uni 4840  df-br 5075  df-opab 5137  df-iota 6391  df-fv 6441

This theorem is referenced by:  fvmptopab  7329  mptmpoopabbrd  7921