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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 6920 . 2 (𝑊𝐺) ∈ V
2 elopabran 5572 . . 3 (𝑧 ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} → 𝑧 ∈ (𝑊𝐺))
32ssriv 3999 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} ⊆ (𝑊𝐺)
41, 3ssexi 5328 1 {⟨𝑥, 𝑦⟩ ∣ (𝑥(𝑊𝐺)𝑦𝜃)} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wa 395  wcel 2106  Vcvv 3478   class class class wbr 5148  {copab 5210  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-sn 4632  df-pr 4634  df-uni 4913  df-br 5149  df-opab 5211  df-iota 6516  df-fv 6571
This theorem is referenced by:  fvmptopab  7487  mptmpoopabbrd  8104  mptmpoopabbrdOLD  8105
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