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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 22076 | . 2 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴)) | |
2 | 1 | simplbi 500 | 1 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 ∩ cin 3934 𝒫 cpw 4538 {csn 4566 ‘cfv 6354 (class class class)co 7155 ↾t crest 16693 Topctop 21500 neicnei 21704 𝑛-Locally cnlly 22072 |
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 |
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-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-iota 6313 df-fv 6362 df-ov 7158 df-nlly 22074 |
This theorem is referenced by: nlly2i 22083 restnlly 22089 nllyrest 22093 nllyidm 22096 cldllycmp 22102 llycmpkgen 22159 txnlly 22244 txkgen 22259 xkococnlem 22266 xkococn 22267 cnmptkk 22290 xkofvcn 22291 cnmptk1p 22292 cnmptk2 22293 xkocnv 22421 xkohmeo 22422 |
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