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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 45792 | . 2 ⊢ (𝑁 ∈ ((nei‘𝐽)‘𝑆) → (nei‘𝐽) ≠ ∅) | |
2 | n0 4247 | . . 3 ⊢ ((nei‘𝐽) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (nei‘𝐽)) | |
3 | 2 | biimpi 219 | . 2 ⊢ ((nei‘𝐽) ≠ ∅ → ∃𝑓 𝑓 ∈ (nei‘𝐽)) |
4 | df-nei 21949 | . . . 4 ⊢ nei = (𝑗 ∈ Top ↦ (𝑥 ∈ 𝒫 ∪ 𝑗 ↦ {𝑦 ∈ 𝒫 ∪ 𝑗 ∣ ∃𝑔 ∈ 𝑗 (𝑥 ⊆ 𝑔 ∧ 𝑔 ⊆ 𝑦)})) | |
5 | 4 | mptrcl 6805 | . . 3 ⊢ (𝑓 ∈ (nei‘𝐽) → 𝐽 ∈ Top) |
6 | 5 | exlimiv 1938 | . 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 399 ∃wex 1787 ∈ wcel 2112 ≠ wne 2932 ∃wrex 3052 {crab 3055 ⊆ wss 3853 ∅c0 4223 𝒫 cpw 4499 ∪ cuni 4805 ↦ cmpt 5120 ‘cfv 6358 Topctop 21744 neicnei 21948 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-xp 5542 df-rel 5543 df-cnv 5544 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6316 df-fv 6366 df-nei 21949 |
This theorem is referenced by: (None) |
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