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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrneineine0lem | Structured version Visualization version GIF version |
Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then conditions equal to claiming that for every point, at least one (pseudo-)neighborbood exists hold equally. (Contributed by RP, 29-May-2021.) |
Ref | Expression |
---|---|
ntrnei.o | ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) |
ntrnei.f | ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) |
ntrnei.r | ⊢ (𝜑 → 𝐼𝐹𝑁) |
ntrnei.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
ntrneineine0lem | ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ (𝑁‘𝑋) ≠ ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrnei.o | . . . 4 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
2 | ntrnei.f | . . . 4 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
3 | ntrnei.r | . . . . 5 ⊢ (𝜑 → 𝐼𝐹𝑁) | |
4 | 3 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁) |
5 | ntrnei.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | 5 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵) |
7 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵) | |
8 | 1, 2, 4, 6, 7 | ntrneiel 44071 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼‘𝑠) ↔ 𝑠 ∈ (𝑁‘𝑋))) |
9 | 8 | rexbidva 3175 | . 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁‘𝑋))) |
10 | 1, 2, 3 | ntrneinex 44067 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) |
11 | elmapi 8888 | . . . . . . . . . 10 ⊢ (𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵) → 𝑁:𝐵⟶𝒫 𝒫 𝐵) | |
12 | 10, 11 | syl 17 | . . . . . . . . 9 ⊢ (𝜑 → 𝑁:𝐵⟶𝒫 𝒫 𝐵) |
13 | 12, 5 | ffvelcdmd 7105 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝒫 𝒫 𝐵) |
14 | 13 | elpwid 4614 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘𝑋) ⊆ 𝒫 𝐵) |
15 | 14 | sseld 3994 | . . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝑁‘𝑋) → 𝑠 ∈ 𝒫 𝐵)) |
16 | 15 | pm4.71rd 562 | . . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝑁‘𝑋) ↔ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑠 ∈ (𝑁‘𝑋)))) |
17 | 16 | exbidv 1919 | . . . 4 ⊢ (𝜑 → (∃𝑠 𝑠 ∈ (𝑁‘𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ 𝑠 ∈ (𝑁‘𝑋)))) |
18 | 17 | bicomd 223 | . . 3 ⊢ (𝜑 → (∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ 𝑠 ∈ (𝑁‘𝑋)) ↔ ∃𝑠 𝑠 ∈ (𝑁‘𝑋))) |
19 | df-rex 3069 | . . 3 ⊢ (∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁‘𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ 𝑠 ∈ (𝑁‘𝑋))) | |
20 | n0 4359 | . . 3 ⊢ ((𝑁‘𝑋) ≠ ∅ ↔ ∃𝑠 𝑠 ∈ (𝑁‘𝑋)) | |
21 | 18, 19, 20 | 3bitr4g 314 | . 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁‘𝑋) ↔ (𝑁‘𝑋) ≠ ∅)) |
22 | 9, 21 | bitrd 279 | 1 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ (𝑁‘𝑋) ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∃wex 1776 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 {crab 3433 Vcvv 3478 ∅c0 4339 𝒫 cpw 4605 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ∈ cmpo 7433 ↑m cmap 8865 |
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-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
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-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 |
This theorem is referenced by: ntrneineine0 44077 |
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