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Mirrors > Home > ILE Home > Th. List > neiss2 | 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 | neifval.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | neif 12349 | . . . . 5 ⊢ (𝐽 ∈ Top → (nei‘𝐽) Fn 𝒫 𝑋) |
3 | fnrel 5229 | . . . . 5 ⊢ ((nei‘𝐽) Fn 𝒫 𝑋 → Rel (nei‘𝐽)) | |
4 | 2, 3 | syl 14 | . . . 4 ⊢ (𝐽 ∈ Top → Rel (nei‘𝐽)) |
5 | relelfvdm 5461 | . . . 4 ⊢ ((Rel (nei‘𝐽) ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ∈ dom (nei‘𝐽)) | |
6 | 4, 5 | sylan 281 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ∈ dom (nei‘𝐽)) |
7 | fndm 5230 | . . . . . 6 ⊢ ((nei‘𝐽) Fn 𝒫 𝑋 → dom (nei‘𝐽) = 𝒫 𝑋) | |
8 | 2, 7 | syl 14 | . . . . 5 ⊢ (𝐽 ∈ Top → dom (nei‘𝐽) = 𝒫 𝑋) |
9 | 8 | eleq2d 2210 | . . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ dom (nei‘𝐽) ↔ 𝑆 ∈ 𝒫 𝑋)) |
10 | 9 | adantr 274 | . . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → (𝑆 ∈ dom (nei‘𝐽) ↔ 𝑆 ∈ 𝒫 𝑋)) |
11 | 6, 10 | mpbid 146 | . 2 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ∈ 𝒫 𝑋) |
12 | 11 | elpwid 3526 | 1 ⊢ ((𝐽 ∈ Top ∧ 𝑁 ∈ ((nei‘𝐽)‘𝑆)) → 𝑆 ⊆ 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∈ wcel 1481 ⊆ wss 3076 𝒫 cpw 3515 ∪ cuni 3744 dom cdm 4547 Rel wrel 4552 Fn wfn 5126 ‘cfv 5131 Topctop 12203 neicnei 12346 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-coll 4051 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-csb 3008 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-iun 3823 df-br 3938 df-opab 3998 df-mpt 3999 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-top 12204 df-nei 12347 |
This theorem is referenced by: neii1 12355 neii2 12357 neiss 12358 ssnei2 12365 topssnei 12370 innei 12371 neitx 12476 |
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