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Theorem neif 22928
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 = βˆͺ 𝐽
Assertion
Ref Expression
neif (𝐽 ∈ Top β†’ (neiβ€˜π½) Fn 𝒫 𝑋)

Proof of Theorem neif
Dummy variables 𝑔 𝑣 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neifval.1 . . . . . 6 𝑋 = βˆͺ 𝐽
21topopn 22732 . . . . 5 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
3 pwexg 5367 . . . . 5 (𝑋 ∈ 𝐽 β†’ 𝒫 𝑋 ∈ V)
4 rabexg 5322 . . . . 5 (𝒫 𝑋 ∈ V β†’ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)} ∈ V)
52, 3, 43syl 18 . . . 4 (𝐽 ∈ Top β†’ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)} ∈ V)
65ralrimivw 3142 . . 3 (𝐽 ∈ Top β†’ βˆ€π‘₯ ∈ 𝒫 𝑋{𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)} ∈ V)
7 eqid 2724 . . . 4 (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)}) = (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)})
87fnmpt 6681 . . 3 (βˆ€π‘₯ ∈ 𝒫 𝑋{𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)} ∈ V β†’ (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)}) Fn 𝒫 𝑋)
96, 8syl 17 . 2 (𝐽 ∈ Top β†’ (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)}) Fn 𝒫 𝑋)
101neifval 22927 . . 3 (𝐽 ∈ Top β†’ (neiβ€˜π½) = (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)}))
1110fneq1d 6633 . 2 (𝐽 ∈ Top β†’ ((neiβ€˜π½) Fn 𝒫 𝑋 ↔ (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)}) Fn 𝒫 𝑋))
129, 11mpbird 257 1 (𝐽 ∈ Top β†’ (neiβ€˜π½) Fn 𝒫 𝑋)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  βˆƒwrex 3062  {crab 3424  Vcvv 3466   βŠ† wss 3941  π’« cpw 4595  βˆͺ cuni 4900   ↦ cmpt 5222   Fn wfn 6529  β€˜cfv 6534  Topctop 22719  neicnei 22925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-top 22720  df-nei 22926
This theorem is referenced by:  neiss2  22929
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