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Theorem opabresex2 7404
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(𝑊𝐺)𝑦𝜃)} ∈ 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.
StepHypRef Expression
1 fvex 6853 . 2 (𝑊𝐺) ∈ V
2 elopabran 5518 . . 3 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} → 𝑧 ∈ (𝑊𝐺))
32ssriv 3947 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} ⊆ (𝑊𝐺)
41, 3ssexi 5278 1 {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 397  wcel 2107  Vcvv 3444   class class class wbr 5104  {copab 5166  cfv 6494
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2709  ax-sep 5255  ax-nul 5262
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2943  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-sn 4586  df-pr 4588  df-uni 4865  df-br 5105  df-opab 5167  df-iota 6446  df-fv 6502
This theorem is referenced by:  fvmptopab  7406  mptmpoopabbrd  8006
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