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Mirrors > Home > MPE Home > Th. List > hausnlly | Structured version Visualization version GIF version |
Description: A Hausdorff space is n-locally Hausdorff iff it is locally Hausdorff (both notions are thus referred to as "locally Hausdorff"). (Contributed by Mario Carneiro, 2-Mar-2015.) |
Ref | Expression |
---|---|
hausnlly | ⊢ (𝐽 ∈ 𝑛-Locally Haus ↔ 𝐽 ∈ Locally Haus) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | resthaus 22112 | . . . . 5 ⊢ ((𝑗 ∈ Haus ∧ 𝑥 ∈ 𝑗) → (𝑗 ↾t 𝑥) ∈ Haus) | |
2 | 1 | adantl 485 | . . . 4 ⊢ ((⊤ ∧ (𝑗 ∈ Haus ∧ 𝑥 ∈ 𝑗)) → (𝑗 ↾t 𝑥) ∈ Haus) |
3 | 2 | restnlly 22226 | . . 3 ⊢ (⊤ → 𝑛-Locally Haus = Locally Haus) |
4 | 3 | mptru 1549 | . 2 ⊢ 𝑛-Locally Haus = Locally Haus |
5 | 4 | eleq2i 2824 | 1 ⊢ (𝐽 ∈ 𝑛-Locally Haus ↔ 𝐽 ∈ Locally Haus) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1542 ⊤wtru 1543 ∈ wcel 2113 (class class class)co 7164 ↾t crest 16790 Hauscha 22052 Locally clly 22208 𝑛-Locally cnlly 22209 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-rep 5151 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3399 df-sbc 3680 df-csb 3789 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-pss 3860 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-tp 4518 df-op 4520 df-uni 4794 df-int 4834 df-iun 4880 df-br 5028 df-opab 5090 df-mpt 5108 df-tr 5134 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6169 df-on 6170 df-lim 6171 df-suc 6172 df-iota 6291 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7167 df-oprab 7168 df-mpo 7169 df-om 7594 df-1st 7707 df-2nd 7708 df-map 8432 df-en 8549 df-fin 8552 df-fi 8941 df-rest 16792 df-topgen 16813 df-top 21638 df-topon 21655 df-bases 21690 df-nei 21842 df-cn 21971 df-haus 22059 df-lly 22210 df-nlly 22211 |
This theorem is referenced by: (None) |
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