Theorem opabresex2 7400

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 6835	. 2 ⊢ (𝑊‘𝐺) ∈ V
2		elopabran 5499	. . 3 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} → 𝑧 ∈ (𝑊‘𝐺))
3	2	ssriv 3933	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ⊆ (𝑊‘𝐺)
4	1, 3	ssexi 5258	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V

Syntax hints:   ∧ wa 395   ∈ wcel 2111  Vcvv 3436   class class class wbr 5089  {copab 5151  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-sn 4574  df-pr 4576  df-uni 4857  df-br 5090  df-opab 5152  df-iota 6437  df-fv 6489

This theorem is referenced by:  fvmptopab  7401  mptmpoopabbrd  8012  mptmpoopabbrdOLD  8013