Theorem neif 14823

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.)

Hypothesis

Ref	Expression
neifval.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
neif	⊢ (𝐽 ∈ Top → (nei‘𝐽) Fn 𝒫 𝑋)

Proof of Theorem neif

Dummy variables 𝑔 𝑣 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		neifval.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
2	1	topopn 14690	. . . . 5 ⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽)
3		pwexg 4264	. . . . 5 ⊢ (𝑋 ∈ 𝐽 → 𝒫 𝑋 ∈ V)
4		rabexg 4227	. . . . 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 2604	. . 3 ⊢ (𝐽 ∈ Top → ∀𝑥 ∈ 𝒫 𝑋{𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)} ∈ V)
7		eqid 2229	. . . 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 5450	. . 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 14822	. . 3 ⊢ (𝐽 ∈ Top → (nei‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔 ∈ 𝐽 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑣)}))
11	10	fneq1d 5411	. 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 167	1 ⊢ (𝐽 ∈ Top → (nei‘𝐽) Fn 𝒫 𝑋)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  {crab 2512  Vcvv 2799   ⊆ wss 3197  𝒫 cpw 3649  ∪ cuni 3888   ↦ cmpt 4145   Fn wfn 5313  ‘cfv 5318  Topctop 14679  neicnei 14820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-coll 4199  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-top 14680  df-nei 14821

This theorem is referenced by:  neiss2  14824