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Theorem xrninxpex 38105
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 7750 . . 3 ((𝐵𝑊𝐶𝑋) → (𝐵 × 𝐶) ∈ V)
2 inxpex 38050 . . . 4 (((𝑅𝑆) ∈ V ∨ (𝐴𝑉 ∧ (𝐵 × 𝐶) ∈ V)) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
32olcs 874 . . 3 ((𝐴𝑉 ∧ (𝐵 × 𝐶) ∈ V) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
41, 3sylan2 591 . 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 394  w3a 1084  wcel 2099  Vcvv 3462  cin 3945   × cxp 5672  cxrn 37888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-opab 5208  df-xp 5680  df-rel 5681
This theorem is referenced by:  xrnresex  38117
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