Theorem xrninxpex 36447

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	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)

Proof of Theorem xrninxpex

Step	Hyp	Ref	Expression
1		xpexg 7578	. . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐵 × 𝐶) ∈ V)
2		inxpex 36401	. . . 4 ⊢ (((𝑅 ⋉ 𝑆) ∈ V ∨ (𝐴 ∈ 𝑉 ∧ (𝐵 × 𝐶) ∈ V)) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
3	2	olcs 872	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 × 𝐶) ∈ V) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
4	1, 3	sylan2 592	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋)) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)
5	4	3impb 1113	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882   × cxp 5578   ⋉ cxrn 36259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-opab 5133  df-xp 5586  df-rel 5587

This theorem is referenced by:  xrnresex  36459