Theorem neircl 48897

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 48841	. 2 ⊢ (𝑁 ∈ ((nei‘𝐽)‘𝑆) → (nei‘𝐽) ≠ ∅)
2		n0 4319	. . 3 ⊢ ((nei‘𝐽) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (nei‘𝐽))
3	2	biimpi 216	. 2 ⊢ ((nei‘𝐽) ≠ ∅ → ∃𝑓 𝑓 ∈ (nei‘𝐽))
4		df-nei 22992	. . . 4 ⊢ nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∈ 𝒫 ∪ 𝑗 ∣ ∃𝑔 ∈ 𝑗 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)}))
5	4	mptrcl 6980	. . 3 ⊢ (𝑓 ∈ (nei‘𝐽) → 𝐽 ∈ Top)
6	5	exlimiv 1930	. 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 1779   ∈ wcel 2109   ≠ wne 2926  ∃wrex 3054  {crab 3408   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  ∪ cuni 4874   ↦ cmpt 5191  ‘cfv 6514  Topctop 22787  neicnei 22991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-xp 5647  df-rel 5648  df-cnv 5649  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fv 6522  df-nei 22992

This theorem is referenced by: (None)