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Theorem xrninxpex 34695
 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 7225 . . 3 ((𝐵𝑊𝐶𝑋) → (𝐵 × 𝐶) ∈ V)
2 inxpex 34650 . . . 4 (((𝑅𝑆) ∈ V ∨ (𝐴𝑉 ∧ (𝐵 × 𝐶) ∈ V)) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
32olcs 907 . . 3 ((𝐴𝑉 ∧ (𝐵 × 𝐶) ∈ V) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
41, 3sylan2 586 . 2 ((𝐴𝑉 ∧ (𝐵𝑊𝐶𝑋)) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
543impb 1147 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ((𝑅𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   ∧ w3a 1111   ∈ wcel 2164  Vcvv 3414   ∩ cin 3797   × cxp 5344   ⋉ cxrn 34518 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-opab 4938  df-xp 5352  df-rel 5353 This theorem is referenced by:  xrnresex  34707
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