Users' Mathboxes Mathbox for Zhi Wang < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  neircl Structured version   Visualization version   GIF version

Theorem neircl 46209
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 46187 . 2 (𝑁 ∈ ((nei‘𝐽)‘𝑆) → (nei‘𝐽) ≠ ∅)
2 n0 4281 . . 3 ((nei‘𝐽) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (nei‘𝐽))
32biimpi 215 . 2 ((nei‘𝐽) ≠ ∅ → ∃𝑓 𝑓 ∈ (nei‘𝐽))
4 df-nei 22258 . . . 4 nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 𝑗 ↦ {𝑦 ∈ 𝒫 𝑗 ∣ ∃𝑔𝑗 (𝑥𝑔𝑔𝑦)}))
54mptrcl 6893 . . 3 (𝑓 ∈ (nei‘𝐽) → 𝐽 ∈ Top)
65exlimiv 1934 . 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 396  wex 1782  wcel 2107  wne 2944  wrex 3066  {crab 3069  wss 3888  c0 4257  𝒫 cpw 4534   cuni 4840  cmpt 5158  cfv 6437  Topctop 22051  neicnei 22257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-sep 5224  ax-nul 5231  ax-pr 5353
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3435  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5076  df-opab 5138  df-mpt 5159  df-xp 5596  df-rel 5597  df-cnv 5598  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6395  df-fv 6445  df-nei 22258
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator