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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 23493 | . 2 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (((nei‘𝐽)‘{𝑦}) ∩ 𝒫 𝑥)(𝐽 ↾t 𝑢) ∈ 𝐴)) | |
2 | 1 | simplbi 497 | 1 ⊢ (𝐽 ∈ 𝑛-Locally 𝐴 → 𝐽 ∈ Top) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 ∩ cin 3962 𝒫 cpw 4605 {csn 4631 ‘cfv 6563 (class class class)co 7431 ↾t crest 17467 Topctop 22915 neicnei 23121 𝑛-Locally cnlly 23489 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-iota 6516 df-fv 6571 df-ov 7434 df-nlly 23491 |
This theorem is referenced by: nlly2i 23500 restnlly 23506 nllyrest 23510 nllyidm 23513 cldllycmp 23519 llycmpkgen 23576 txnlly 23661 txkgen 23676 xkococnlem 23683 xkococn 23684 cnmptkk 23707 xkofvcn 23708 cnmptk1p 23709 cnmptk2 23710 xkocnv 23838 xkohmeo 23839 |
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