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Theorem neircl 47699
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 47677 . 2 (𝑁 ∈ ((neiβ€˜π½)β€˜π‘†) β†’ (neiβ€˜π½) β‰  βˆ…)
2 n0 4346 . . 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 22922 . . . 4 nei = (𝑗 ∈ Top ↦ (π‘₯ ∈ 𝒫 βˆͺ 𝑗 ↦ {𝑦 ∈ 𝒫 βˆͺ 𝑗 ∣ βˆƒπ‘” ∈ 𝑗 (π‘₯ βŠ† 𝑔 ∧ 𝑔 βŠ† 𝑦)}))
54mptrcl 7007 . . 3 (𝑓 ∈ (neiβ€˜π½) β†’ 𝐽 ∈ Top)
65exlimiv 1932 . 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 395  βˆƒwex 1780   ∈ wcel 2105   β‰  wne 2939  βˆƒwrex 3069  {crab 3431   βŠ† wss 3948  βˆ…c0 4322  π’« cpw 4602  βˆͺ cuni 4908   ↦ cmpt 5231  β€˜cfv 6543  Topctop 22715  neicnei 22921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5682  df-rel 5683  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fv 6551  df-nei 22922
This theorem is referenced by: (None)
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