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Mirrors > Home > MPE Home > Th. List > neiss2 | Structured version Visualization version GIF version |
Description: A set with a neighborhood is a subset of the base set of a topology. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by NM, 12-Feb-2007.) |
Ref | Expression |
---|---|
neifval.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
neiss2 | ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 6943 | . . . 4 ⊢ (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝑆 ∈ dom (nei‘𝐽)) | |
2 | 1 | adantl 481 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ∈ dom (nei‘𝐽)) |
3 | neifval.1 | . . . . . . 7 ⊢ 𝑋 = ∪ 𝐽 | |
4 | 3 | neif 23123 | . . . . . 6 ⊢ (𝐽 ∈ Top → (nei‘𝐽) Fn 𝒫 𝑋) |
5 | 4 | fndmd 6673 | . . . . 5 ⊢ (𝐽 ∈ Top → dom (nei‘𝐽) = 𝒫 𝑋) |
6 | 5 | eleq2d 2824 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ dom (nei‘𝐽) ↔ 𝑆 ∈ 𝒫 𝑋)) |
7 | 6 | adantr 480 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑆 ∈ dom (nei‘𝐽) ↔ 𝑆 ∈ 𝒫 𝑋)) |
8 | 2, 7 | mpbid 232 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ∈ 𝒫 𝑋) |
9 | 8 | elpwid 4613 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ⊆ wss 3962 𝒫 cpw 4604 ∪ cuni 4911 dom cdm 5688 ‘cfv 6562 Topctop 22914 neicnei 23120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-top 22915 df-nei 23121 |
This theorem is referenced by: neii1 23129 neii2 23131 neiss 23132 ssnei2 23139 topssnei 23147 innei 23148 neitx 23630 cvmlift2lem12 35298 neiin 36314 cnneiima 48712 |
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