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Mirrors > Home > MPE Home > Th. List > Mathboxes > neircl | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
neircl | ⊢ (𝑁 ∈ ((nei‘𝐽)‘𝑆) → 𝐽 ∈ Top) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvne0 48679 | . 2 ⊢ (𝑁 ∈ ((nei‘𝐽)‘𝑆) → (nei‘𝐽) ≠ ∅) | |
2 | n0 4359 | . . 3 ⊢ ((nei‘𝐽) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (nei‘𝐽)) | |
3 | 2 | biimpi 216 | . 2 ⊢ ((nei‘𝐽) ≠ ∅ → ∃𝑓 𝑓 ∈ (nei‘𝐽)) |
4 | df-nei 23122 | . . . 4 ⊢ nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∈ 𝒫 ∪ 𝑗 ∣ ∃𝑔 ∈ 𝑗 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)})) | |
5 | 4 | mptrcl 7025 | . . 3 ⊢ (𝑓 ∈ (nei‘𝐽) → 𝐽 ∈ Top) |
6 | 5 | exlimiv 1928 | . 2 ⊢ (∃𝑓 𝑓 ∈ (nei‘𝐽) → 𝐽 ∈ Top) |
7 | 1, 3, 6 | 3syl 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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