Theorem neircl 49395

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.

Step	Hyp	Ref	Expression
1		elfvne0 49339	. 2 ⊢ (𝑁 ∈ ((nei‘𝐽)‘𝑆) → (nei‘𝐽) ≠ ∅)
2		n0 4281	. . 3 ⊢ ((nei‘𝐽) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (nei‘𝐽))
3	2	biimpi 217	. 2 ⊢ ((nei‘𝐽) ≠ ∅ → ∃𝑓 𝑓 ∈ (nei‘𝐽))
4		df-nei 23081	. . . 4 ⊢ nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∈ 𝒫 ∪ 𝑗 ∣ ∃𝑔 ∈ 𝑗 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)}))
5	4	mptrcl 6945	. . 3 ⊢ (𝑓 ∈ (nei‘𝐽) → 𝐽 ∈ Top)
6	5	exlimiv 1937	. 2 ⊢ (∃𝑓 𝑓 ∈ (nei‘𝐽) → 𝐽 ∈ Top)
7	1, 3, 6	3syl 18	1 ⊢ (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∧ wa 396  ∃wex 1786   ∈ wcel 2119   ≠ wne 2934  ∃wrex 3063  {crab 3391   ⊆ wss 3883  ∅c0 4261  𝒫 cpw 4529  ∪ cuni 4838   ↦ cmpt 5153  ‘cfv 6485  Topctop 22876  neicnei 23080

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-xp 5624  df-rel 5625  df-cnv 5626  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fv 6493  df-nei 23081

This theorem is referenced by: (None)