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Mirrors > Home > MPE Home > Th. List > mrerintcl | Structured version Visualization version GIF version |
Description: The relative intersection of a set of closed sets is closed. (Contributed by Stefan O'Rear, 3-Apr-2015.) |
Ref | Expression |
---|---|
mrerintcl | ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rint0 4908 | . . . 4 ⊢ (𝑆 = ∅ → (𝑋 ∩ ∩ 𝑆) = 𝑋) | |
2 | 1 | adantl 484 | . . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 = ∅) → (𝑋 ∩ ∩ 𝑆) = 𝑋) |
3 | mre1cl 16859 | . . . 4 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ 𝐶) | |
4 | 3 | ad2antrr 724 | . . 3 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 = ∅) → 𝑋 ∈ 𝐶) |
5 | 2, 4 | eqeltrd 2913 | . 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 = ∅) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶) |
6 | simp2 1133 | . . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ 𝐶) | |
7 | mresspw 16857 | . . . . . . 7 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐶 ⊆ 𝒫 𝑋) | |
8 | 7 | 3ad2ant1 1129 | . . . . . 6 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝐶 ⊆ 𝒫 𝑋) |
9 | 6, 8 | sstrd 3976 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ⊆ 𝒫 𝑋) |
10 | simp3 1134 | . . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → 𝑆 ≠ ∅) | |
11 | rintn0 5022 | . . . . 5 ⊢ ((𝑆 ⊆ 𝒫 𝑋 ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) = ∩ 𝑆) | |
12 | 9, 10, 11 | syl2anc 586 | . . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) = ∩ 𝑆) |
13 | mreintcl 16860 | . . . 4 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → ∩ 𝑆 ∈ 𝐶) | |
14 | 12, 13 | eqeltrd 2913 | . . 3 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶 ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶) |
15 | 14 | 3expa 1114 | . 2 ⊢ (((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) ∧ 𝑆 ≠ ∅) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶) |
16 | 5, 15 | pm2.61dane 3104 | 1 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ⊆ 𝐶) → (𝑋 ∩ ∩ 𝑆) ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 ∩ cin 3934 ⊆ wss 3935 ∅c0 4290 𝒫 cpw 4538 ∩ cint 4868 ‘cfv 6349 Moorecmre 16847 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-int 4869 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-iota 6308 df-fun 6351 df-fv 6357 df-mre 16851 |
This theorem is referenced by: mreacs 16923 topmtcl 33706 |
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