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Theorem opabex2 7990
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 7686 . 2 (𝜑 → (𝐴 × 𝐵) ∈ V)
4 opabex2.3 . . 3 ((𝜑𝜓) → 𝑥𝐴)
5 opabex2.4 . . 3 ((𝜑𝜓) → 𝑦𝐵)
64, 5opabssxpd 5680 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))
73, 6ssexd 5282 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wcel 2107  Vcvv 3446  {copab 5168   × cxp 5632
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 2708  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-opab 5169  df-xp 5640  df-rel 5641
This theorem is referenced by:  tgjustf  27418  legval  27529  wksvOLD  28571  mgcoval  31849  satf00  33971  bj-imdirval2lem  35656  rfovcnvfvd  42286  sprsymrelfvlem  45689
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