Theorem neircl 49264

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 49208	. 2 ⊢ (𝑁 ∈ ((nei‘𝐽)‘𝑆) → (nei‘𝐽) ≠ ∅)
2		n0 4307	. . 3 ⊢ ((nei‘𝐽) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (nei‘𝐽))
3	2	biimpi 216	. 2 ⊢ ((nei‘𝐽) ≠ ∅ → ∃𝑓 𝑓 ∈ (nei‘𝐽))
4		df-nei 23054	. . . 4 ⊢ nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∈ 𝒫 ∪ 𝑗 ∣ ∃𝑔 ∈ 𝑗 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)}))
5	4	mptrcl 6959	. . 3 ⊢ (𝑓 ∈ (nei‘𝐽) → 𝐽 ∈ Top)
6	5	exlimiv 1932	. 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 395  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  ∃wrex 3062  {crab 3401   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  ∪ cuni 4865   ↦ cmpt 5181  ‘cfv 6500  Topctop 22849  neicnei 23053

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fv 6508  df-nei 23054

This theorem is referenced by: (None)