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Theorem xrninxpex 36262
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 7540 . . 3 ((𝐵𝑊𝐶𝑋) → (𝐵 × 𝐶) ∈ V)
2 inxpex 36216 . . . 4 (((𝑅𝑆) ∈ V ∨ (𝐴𝑉 ∧ (𝐵 × 𝐶) ∈ V)) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
32olcs 876 . . 3 ((𝐴𝑉 ∧ (𝐵 × 𝐶) ∈ V) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
41, 3sylan2 596 . 2 ((𝐴𝑉 ∧ (𝐵𝑊𝐶𝑋)) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
543impb 1117 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089  wcel 2110  Vcvv 3413  cin 3870   × cxp 5554  cxrn 36074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2708  ax-sep 5197  ax-nul 5204  ax-pow 5263  ax-pr 5327  ax-un 7528
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3415  df-dif 3874  df-un 3876  df-in 3878  df-ss 3888  df-nul 4243  df-if 4445  df-pw 4520  df-sn 4547  df-pr 4549  df-op 4553  df-uni 4825  df-opab 5121  df-xp 5562  df-rel 5563
This theorem is referenced by:  xrnresex  36274
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