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Theorem xrninxpex 35634
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 7465 . . 3 ((𝐵𝑊𝐶𝑋) → (𝐵 × 𝐶) ∈ V)
2 inxpex 35588 . . . 4 (((𝑅𝑆) ∈ V ∨ (𝐴𝑉 ∧ (𝐵 × 𝐶) ∈ V)) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
32olcs 872 . . 3 ((𝐴𝑉 ∧ (𝐵 × 𝐶) ∈ V) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
41, 3sylan2 594 . 2 ((𝐴𝑉 ∧ (𝐵𝑊𝐶𝑋)) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
543impb 1110 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1082  wcel 2108  Vcvv 3493  cin 3933   × cxp 5546  cxrn 35444
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 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-opab 5120  df-xp 5554  df-rel 5555
This theorem is referenced by:  xrnresex  35646
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