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Theorem neif 22997
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 22801 . . . . 5 (𝐽 ∈ Top β†’ 𝑋 ∈ 𝐽)
3 pwexg 5372 . . . . 5 (𝑋 ∈ 𝐽 β†’ 𝒫 𝑋 ∈ V)
4 rabexg 5327 . . . . 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 3146 . . 3 (𝐽 ∈ Top β†’ βˆ€π‘₯ ∈ 𝒫 𝑋{𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)} ∈ V)
7 eqid 2728 . . . 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 6689 . . 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 22996 . . 3 (𝐽 ∈ Top β†’ (neiβ€˜π½) = (π‘₯ ∈ 𝒫 𝑋 ↦ {𝑣 ∈ 𝒫 𝑋 ∣ βˆƒπ‘” ∈ 𝐽 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑣)}))
1110fneq1d 6641 . 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 1534   ∈ wcel 2099  βˆ€wral 3057  βˆƒwrex 3066  {crab 3428  Vcvv 3470   βŠ† wss 3945  π’« cpw 4598  βˆͺ cuni 4903   ↦ cmpt 5225   Fn wfn 6537  β€˜cfv 6542  Topctop 22788  neicnei 22994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-top 22789  df-nei 22995
This theorem is referenced by:  neiss2  22998
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