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Theorem neircl 45767
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 ∈ ((nei‘𝐽)‘𝑆) → 𝐽 ∈ Top)

Proof of Theorem neircl
Dummy variables 𝑓 𝑔 𝑗 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvne0 45752 . 2 (𝑁 ∈ ((nei‘𝐽)‘𝑆) → (nei‘𝐽) ≠ ∅)
2 n0 4245 . . 3 ((nei‘𝐽) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (nei‘𝐽))
32biimpi 219 . 2 ((nei‘𝐽) ≠ ∅ → ∃𝑓 𝑓 ∈ (nei‘𝐽))
4 df-nei 21861 . . . 4 nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 ↦ {𝑦 ∈ 𝒫 𝑗 ∣ ∃𝑔𝑗 (𝑥𝑔𝑔𝑦)}))
54mptrcl 6796 . . 3 (𝑓 ∈ (nei‘𝐽) → 𝐽 ∈ Top)
65exlimiv 1937 . 2 (∃𝑓 𝑓 ∈ (nei‘𝐽) → 𝐽 ∈ Top)
71, 3, 63syl 18 1 (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wex 1786  wcel 2114  wne 2935  wrex 3055  {crab 3058  wss 3853  c0 4221  𝒫 cpw 4498   cuni 4806  cmpt 5120  cfv 6349  Topctop 21656  neicnei 21860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-mpt 5121  df-xp 5541  df-rel 5542  df-cnv 5543  df-dm 5545  df-rn 5546  df-res 5547  df-ima 5548  df-iota 6307  df-fv 6357  df-nei 21861
This theorem is referenced by: (None)
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