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Theorem opabex2 8039
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 7734 . 2 (𝜑 → (𝐴 × 𝐵) ∈ V)
4 opabex2.3 . . 3 ((𝜑𝜓) → 𝑥𝐴)
5 opabex2.4 . . 3 ((𝜑𝜓) → 𝑦𝐵)
64, 5opabssxpd 5716 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))
73, 6ssexd 5317 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  Vcvv 3468  {copab 5203   × cxp 5667
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-opab 5204  df-xp 5675  df-rel 5676
This theorem is referenced by:  tgjustf  28227  legval  28338  wksvOLD  29381  mgcoval  32658  satf00  34892  bj-imdirval2lem  36569  rfovcnvfvd  43316  sprsymrelfvlem  46712
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