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Theorem opabresex2 7485
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 6919 . 2 (𝑊𝐺) ∈ V
2 elopabran 5567 . . 3 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} → 𝑧 ∈ (𝑊𝐺))
32ssriv 3987 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} ⊆ (𝑊𝐺)
41, 3ssexi 5322 1 {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 395  wcel 2108  Vcvv 3480   class class class wbr 5143  {copab 5205  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-sn 4627  df-pr 4629  df-uni 4908  df-br 5144  df-opab 5206  df-iota 6514  df-fv 6569
This theorem is referenced by:  fvmptopab  7487  mptmpoopabbrd  8105  mptmpoopabbrdOLD  8106
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