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Theorem neircl 48001
Description: Reverse closure of the neighborhood operation. (This theorem depends on a function's value being empty outside of its domain, but it will make later theorems simpler to state.) (Contributed by Zhi Wang, 16-Sep-2024.)
Assertion
Ref Expression
neircl (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ 𝐽 ∈ Top)

Proof of Theorem neircl
Dummy variables 𝑓 𝑔 𝑗 π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvne0 47979 . 2 (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ (neiβ€˜π½) β‰  βˆ…)
2 n0 4350 . . 3 ((neiβ€˜π½) β‰  βˆ… ↔ βˆƒπ‘“ 𝑓 ∈ (neiβ€˜π½))
32biimpi 215 . 2 ((neiβ€˜π½) β‰  βˆ… β†’ βˆƒπ‘“ 𝑓 ∈ (neiβ€˜π½))
4 df-nei 23022 . . . 4 nei = (𝑗 ∈ Top ↦ (π‘₯ ∈ 𝒫 βˆͺ 𝑗 ↦ {𝑦 ∈ 𝒫 βˆͺ 𝑗 ∣ βˆƒπ‘” ∈ 𝑗 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑦)}))
54mptrcl 7019 . . 3 (𝑓 ∈ (neiβ€˜π½) β†’ 𝐽 ∈ Top)
65exlimiv 1925 . 2 (βˆƒπ‘“ 𝑓 ∈ (neiβ€˜π½) β†’ 𝐽 ∈ Top)
71, 3, 63syl 18 1 (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394  βˆƒwex 1773   ∈ wcel 2098   β‰  wne 2937  βˆƒwrex 3067  {crab 3430   βŠ† wss 3949  βˆ…c0 4326  π’« cpw 4606  βˆͺ cuni 4912   ↦ cmpt 5235  β€˜cfv 6553  Topctop 22815  neicnei 23021
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fv 6561  df-nei 23022
This theorem is referenced by: (None)
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