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Theorem neif 21126
Description: The neighborhood function is a function of the subsets of a topology's base set. (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.
StepHypRef Expression
1 neifval.1 . . . . . 6 𝑋 = 𝐽
21topopn 20933 . . . . 5 (𝐽 ∈ Top → 𝑋𝐽)
3 pwexg 4999 . . . . 5 (𝑋𝐽 → 𝒫 𝑋 ∈ V)
4 rabexg 4963 . . . . 5 (𝒫 𝑋 ∈ V → {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)} ∈ V)
52, 3, 43syl 18 . . . 4 (𝐽 ∈ Top → {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)} ∈ V)
65ralrimivw 3105 . . 3 (𝐽 ∈ Top → ∀𝑥 ∈ 𝒫 𝑋{𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)} ∈ V)
7 eqid 2760 . . . 4 (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)}) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)})
87fnmpt 6181 . . 3 (∀𝑥 ∈ 𝒫 𝑋{𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)} ∈ V → (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)}) Fn 𝒫 𝑋)
96, 8syl 17 . 2 (𝐽 ∈ Top → (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)}) Fn 𝒫 𝑋)
101neifval 21125 . . 3 (𝐽 ∈ Top → (nei‘𝐽) = (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)}))
1110fneq1d 6142 . 2 (𝐽 ∈ Top → ((nei‘𝐽) Fn 𝒫 𝑋 ↔ (𝑥 ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ ∃𝑔𝐽 (𝑥𝑔𝑔𝑣)}) Fn 𝒫 𝑋))
129, 11mpbird 247 1 (𝐽 ∈ Top → (nei‘𝐽) Fn 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  wcel 2139  wral 3050  wrex 3051  {crab 3054  Vcvv 3340  wss 3715  𝒫 cpw 4302   cuni 4588  cmpt 4881   Fn wfn 6044  cfv 6049  Topctop 20920  neicnei 21123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-top 20921  df-nei 21124
This theorem is referenced by:  neiss2  21127
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