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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 46187 | . 2 ⊢ (𝑁 ∈ ((nei‘𝐽)‘𝑆) → (nei‘𝐽) ≠ ∅) | |
2 | n0 4281 | . . 3 ⊢ ((nei‘𝐽) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (nei‘𝐽)) | |
3 | 2 | biimpi 215 | . 2 ⊢ ((nei‘𝐽) ≠ ∅ → ∃𝑓 𝑓 ∈ (nei‘𝐽)) |
4 | df-nei 22258 | . . . 4 ⊢ nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∈ 𝒫 ∪ 𝑗 ∣ ∃𝑔 ∈ 𝑗 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)})) | |
5 | 4 | mptrcl 6893 | . . 3 ⊢ (𝑓 ∈ (nei‘𝐽) → 𝐽 ∈ Top) |
6 | 5 | exlimiv 1934 | . 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 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) |
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