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| Mirrors > Home > MPE Home > Th. List > llytop | Structured version Visualization version GIF version | ||
| Description: A locally 𝐴 space is a topological space. (Contributed by Mario Carneiro, 2-Mar-2015.) | 
| Ref | Expression | 
|---|---|
| llytop | ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | islly 23476 | . 2 ⊢ (𝐽 ∈ Locally 𝐴 ↔ (𝐽 ∈ Top ∧ ∀𝑥 ∈ 𝐽 ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ (𝐽 ∩ 𝒫 𝑥)(𝑦 ∈ 𝑢 ∧ (𝐽 ↾t 𝑢) ∈ 𝐴))) | |
| 2 | 1 | simplbi 497 | 1 ⊢ (𝐽 ∈ Locally 𝐴 → 𝐽 ∈ Top) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ∩ cin 3950 𝒫 cpw 4600 (class class class)co 7431 ↾t crest 17465 Topctop 22899 Locally clly 23472 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-lly 23474 | 
| This theorem is referenced by: llynlly 23485 islly2 23492 llyrest 23493 llyidm 23496 nllyidm 23497 toplly 23498 lly1stc 23504 txlly 23644 | 
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