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Theorem xrninxpex 37777
Description: Sufficient condition for the intersection of a range Cartesian product with a Cartesian product to be a set. (Contributed by Peter Mazsa, 12-Apr-2020.)
Assertion
Ref Expression
xrninxpex ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)

Proof of Theorem xrninxpex
StepHypRef Expression
1 xpexg 7734 . . 3 ((𝐵𝑊𝐶𝑋) → (𝐵 × 𝐶) ∈ V)
2 inxpex 37721 . . . 4 (((𝑅𝑆) ∈ V ∨ (𝐴𝑉 ∧ (𝐵 × 𝐶) ∈ V)) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
32olcs 873 . . 3 ((𝐴𝑉 ∧ (𝐵 × 𝐶) ∈ V) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
41, 3sylan2 592 . 2 ((𝐴𝑉 ∧ (𝐵𝑊𝐶𝑋)) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
543impb 1112 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084  wcel 2098  Vcvv 3468  cin 3942   × cxp 5667  cxrn 37555
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 7722
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:  xrnresex  37789
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