Theorem opabresex2 7477

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 6914	. 2 ⊢ (𝑊‘𝐺) ∈ V
2		elopabran 5568	. . 3 ⊢ (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} → 𝑧 ∈ (𝑊‘𝐺))
3	2	ssriv 3983	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ⊆ (𝑊‘𝐺)
4	1, 3	ssexi 5327	1 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊‘𝐺)𝑦 ∧ 𝜃)} ∈ V

Syntax hints:   ∧ wa 394   ∈ wcel 2099  Vcvv 3462   class class class wbr 5153  {copab 5215  ‘cfv 6554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-sn 4634  df-pr 4636  df-uni 4914  df-br 5154  df-opab 5216  df-iota 6506  df-fv 6562

This theorem is referenced by:  fvmptopab  7479  mptmpoopabbrd  8094  mptmpoopabbrdOLD  8095