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Mirrors > Home > ILE Home > Th. List > neif | GIF version |
Description: The neighborhood function is a function from the set of the subsets of the base set of a topology. (Contributed by NM, 12-Feb-2007.) (Revised by Mario Carneiro, 11-Nov-2013.) |
Ref | Expression |
---|---|
neifval.1 | ⊢ 𝑋 = ∪ 𝐽 |
Ref | Expression |
---|---|
neif | ⊢ (𝐽 ∈ Top → (nei‘𝐽) Fn 𝒫 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | neifval.1 | . . . . . 6 ⊢ 𝑋 = ∪ 𝐽 | |
2 | 1 | topopn 12178 | . . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
3 | pwexg 4104 | . . . . 5 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V) | |
4 | rabexg 4071 | . . . . 5 ⊢ (𝒫 𝑋 ∈ V → {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)} ∈ V) | |
5 | 2, 3, 4 | 3syl 17 | . . . 4 ⊢ (𝐽 ∈ Top → {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)} ∈ V) |
6 | 5 | ralrimivw 2506 | . . 3 ⊢ (𝐽 ∈ Top → ∀𝑥 ∈ 𝒫 𝑋{𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)} ∈ V) |
7 | eqid 2139 | . . . 4 ⊢ (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)}) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)}) | |
8 | 7 | fnmpt 5249 | . . 3 ⊢ (∀𝑥 ∈ 𝒫 𝑋{𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)} ∈ V → (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)}) Fn 𝒫 𝑋) |
9 | 6, 8 | syl 14 | . 2 ⊢ (𝐽 ∈ Top → (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)}) Fn 𝒫 𝑋) |
10 | 1 | neifval 12312 | . . 3 ⊢ (𝐽 ∈ Top → (nei‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)})) |
11 | 10 | fneq1d 5213 | . 2 ⊢ (𝐽 ∈ Top → ((nei‘𝐽) Fn 𝒫 𝑋 ↔ (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)}) Fn 𝒫 𝑋)) |
12 | 9, 11 | mpbird 166 | 1 ⊢ (𝐽 ∈ Top → (nei‘𝐽) Fn 𝒫 𝑋) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1331 ∈ wcel 1480 ∀wral 2416 ∃wrex 2417 {crab 2420 Vcvv 2686 ⊆ wss 3071 𝒫 cpw 3510 ∪ cuni 3736 ↦ cmpt 3989 Fn wfn 5118 ‘cfv 5123 Topctop 12167 neicnei 12310 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-coll 4043 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-top 12168 df-nei 12311 |
This theorem is referenced by: neiss2 12314 |
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