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Mirrors > Home > MPE Home > Th. List > Mathboxes > xrninxpex | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
xrninxpex | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpexg 7225 | . . 3 ⊢ ((𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝐵 × 𝐶) ∈ V) | |
2 | inxpex 34650 | . . . 4 ⊢ (((𝑅 ⋉ 𝑆) ∈ V ∨ (𝐴 ∈ 𝑉 ∧ (𝐵 × 𝐶) ∈ V)) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V) | |
3 | 2 | olcs 907 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 × 𝐶) ∈ V) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V) |
4 | 1, 3 | sylan2 586 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋)) → ((𝑅 ⋉ 𝑆) ∩ (𝐴 × (𝐵 × 𝐶))) ∈ V) |
5 | 4 | 3impb 1147 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → ((𝑅 ⋉ 𝑆) ∩ (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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