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Theorem opabex2 7750
Description: Condition for an operation to be a set. (Contributed by Thierry Arnoux, 25-Jun-2019.)
Hypotheses
Ref Expression
opabex2.1 (𝜑𝐴𝑉)
opabex2.2 (𝜑𝐵𝑊)
opabex2.3 ((𝜑𝜓) → 𝑥𝐴)
opabex2.4 ((𝜑𝜓) → 𝑦𝐵)
Assertion
Ref Expression
opabex2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem opabex2
StepHypRef Expression
1 opabex2.1 . . 3 (𝜑𝐴𝑉)
2 opabex2.2 . . 3 (𝜑𝐵𝑊)
31, 2xpexd 7468 . 2 (𝜑 → (𝐴 × 𝐵) ∈ V)
4 opabex2.3 . . 3 ((𝜑𝜓) → 𝑥𝐴)
5 opabex2.4 . . 3 ((𝜑𝜓) → 𝑦𝐵)
64, 5opabssxpd 5776 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))
73, 6ssexd 5214 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wcel 2115  Vcvv 3480  {copab 5114   × cxp 5540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-opab 5115  df-xp 5548  df-rel 5549
This theorem is referenced by:  tgjustf  26274  legval  26385  wksv  27416  mgcoval  30683  satf00  32682  bj-imdirval2lem  34546  rfovcnvfvd  40630  sprsymrelfvlem  43938
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