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Theorem neircl 48701
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 48679 . 2 (𝑁 ∈ ((nei‘𝐽)‘𝑆) → (nei‘𝐽) ≠ ∅)
2 n0 4359 . . 3 ((nei‘𝐽) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (nei‘𝐽))
32biimpi 216 . 2 ((nei‘𝐽) ≠ ∅ → ∃𝑓 𝑓 ∈ (nei‘𝐽))
4 df-nei 23122 . . . 4 nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 ↦ {𝑦 ∈ 𝒫 𝑗 ∣ ∃𝑔𝑗 (𝑥𝑔𝑔𝑦)}))
54mptrcl 7025 . . 3 (𝑓 ∈ (nei‘𝐽) → 𝐽 ∈ Top)
65exlimiv 1928 . 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 395  wex 1776  wcel 2106  wne 2938  wrex 3068  {crab 3433  wss 3963  c0 4339  𝒫 cpw 4605   cuni 4912  cmpt 5231  cfv 6563  Topctop 22915  neicnei 23121
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fv 6571  df-nei 23122
This theorem is referenced by: (None)
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