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Mirrors > Home > MPE Home > Th. List > tpnei | Structured version Visualization version GIF version |
Description: The underlying set of a topology is a neighborhood of any of its subsets. Special case of opnneiss 21723. (Contributed by FL, 2-Oct-2006.) |
Ref | Expression |
---|---|
tpnei.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
tpnei | ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tpnei.1 | . . . 4 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | topopn 21511 | . . 3 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
3 | opnneiss 21723 | . . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ∈ 𝐽 ∧ 𝑆 ⊆ 𝑋) → 𝑋 ∈ ((nei‘𝐽)‘𝑆)) | |
4 | 3 | 3exp 1116 | . . 3 ⊢ (𝐽 ∈ Top → (𝑋 ∈ 𝐽 → (𝑆 ⊆ 𝑋 → 𝑋 ∈ ((nei‘𝐽)‘𝑆)))) |
5 | 2, 4 | mpd 15 | . 2 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 → 𝑋 ∈ ((nei‘𝐽)‘𝑆))) |
6 | ssnei 21715 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑋 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋) | |
7 | 6 | ex 416 | . 2 ⊢ (𝐽 ∈ Top → (𝑋 ∈ ((nei‘𝐽)‘𝑆) → 𝑆 ⊆ 𝑋)) |
8 | 5, 7 | impbid 215 | 1 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ 𝑋 ↔ 𝑋 ∈ ((nei‘𝐽)‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 ∪ cuni 4800 ‘cfv 6324 Topctop 21498 neicnei 21702 |
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 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
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-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 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-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-top 21499 df-nei 21703 |
This theorem is referenced by: neiuni 21727 neifil 22485 gneispa 40833 |
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