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Theorem opabex2 7376
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)
3 xpexg 7107 . . 3 ((𝐴𝑉𝐵𝑊) → (𝐴 × 𝐵) ∈ V)
41, 2, 3syl2anc 573 . 2 (𝜑 → (𝐴 × 𝐵) ∈ V)
5 opabex2.3 . . 3 ((𝜑𝜓) → 𝑥𝐴)
6 opabex2.4 . . 3 ((𝜑𝜓) → 𝑦𝐵)
75, 6opabssxpd 5476 . 2 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ (𝐴 × 𝐵))
84, 7ssexd 4939 1 (𝜑 → {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 2145  Vcvv 3351  {copab 4846   × cxp 5247
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-opab 4847  df-xp 5255  df-rel 5256
This theorem is referenced by:  legval  25700  wksv  26750  rfovcnvfvd  38827  sprsymrelfvlem  42268
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