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Mirrors > Home > MPE Home > Th. List > nllytop | Structured version Visualization version GIF version |
Description: A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) |
Ref | Expression |
---|---|
nllytop | ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isnlly 22074 | . 2 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴)) | |
2 | 1 | simplbi 501 | 1 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ∀wral 3106 ∃wrex 3107 ∩ cin 3880 𝒫 cpw 4497 {csn 4525 ‘cfv 6324 (class class class)co 7135 ↾t crest 16686 Topctop 21498 neicnei 21702 𝑛-Locally cnlly 22070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-nlly 22072 |
This theorem is referenced by: nlly2i 22081 restnlly 22087 nllyrest 22091 nllyidm 22094 cldllycmp 22100 llycmpkgen 22157 txnlly 22242 txkgen 22257 xkococnlem 22264 xkococn 22265 cnmptkk 22288 xkofvcn 22289 cnmptk1p 22290 cnmptk2 22291 xkocnv 22419 xkohmeo 22420 |
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