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Theorem inxpex 38341
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 5318 . 2 (𝑅𝑊 → (𝑅 ∩ (𝐴 × 𝐵)) ∈ V)
2 xpexg 7771 . . 3 ((𝐴𝑈𝐵𝑉) → (𝐴 × 𝐵) ∈ V)
3 inex2g 5319 . . 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 2107  Vcvv 3479  cin 3949   × cxp 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-opab 5205  df-xp 5690  df-rel 5691
This theorem is referenced by:  xrninxpex  38396
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