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Theorem inxpex 38321
Description: Sufficient condition for an intersection with a Cartesian product to be a set. (Contributed by Peter Mazsa, 10-May-2019.)
Assertion
Ref Expression
inxpex ((𝑅𝑊 ∨ (𝐴𝑈𝐵𝑉)) → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)

Proof of Theorem inxpex
StepHypRef Expression
1 inex1g 5325 . 2 (𝑅𝑊 → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
2 xpexg 7769 . . 3 ((𝐴𝑈𝐵𝑉) → (𝐴 × 𝐵) ∈ V)
3 inex2g 5326 . . 3 ((𝐴 × 𝐵) ∈ V → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
42, 3syl 17 . 2 ((𝐴𝑈𝐵𝑉) → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
51, 4jaoi 857 1 ((𝑅𝑊 ∨ (𝐴𝑈𝐵𝑉)) → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 847  wcel 2106  Vcvv 3478  cin 3962   × cxp 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-opab 5211  df-xp 5695  df-rel 5696
This theorem is referenced by:  xrninxpex  38376
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