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Theorem opabex2 7744
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 7463 . 2 (𝜑 → (𝐴 × 𝐵) ∈ V)
4 opabex2.3 . . 3 ((𝜑𝜓) → 𝑥𝐴)
5 opabex2.4 . . 3 ((𝜑𝜓) → 𝑦𝐵)
64, 5opabssxpd 5782 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))
73, 6ssexd 5219 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wcel 2105  Vcvv 3492  {copab 5119   × cxp 5546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-opab 5120  df-xp 5554  df-rel 5555
This theorem is referenced by:  tgjustf  26186  legval  26297  wksv  27328  satf00  32518  bj-imdirval2  34365  rfovcnvfvd  40231  sprsymrelfvlem  43529
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