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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 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁) |
5 | ntrnei.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | 5 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵) |
7 | simpr 487 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵) | |
8 | 1, 2, 4, 6, 7 | ntrneiel 40438 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼‘𝑠) ↔ 𝑠 ∈ (𝑁‘𝑋))) |
9 | 8 | rexbidva 3298 | . 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁‘𝑋))) |
10 | 1, 2, 3 | ntrneinex 40434 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (𝒫 𝒫 𝐵 ↑m 𝐵)) |
11 | elmapi 8430 | . . . . . . . . . 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 | ffvelrnd 6854 | . . . . . . . 8 ⊢ (𝜑 → (𝑁‘𝑋) ∈ 𝒫 𝒫 𝐵) |
14 | 13 | elpwid 4552 | . . . . . . 7 ⊢ (𝜑 → (𝑁‘𝑋) ⊆ 𝒫 𝐵) |
15 | 14 | sseld 3968 | . . . . . 6 ⊢ (𝜑 → (𝑠 ∈ (𝑁‘𝑋) → 𝑠 ∈ 𝒫 𝐵)) |
16 | 15 | pm4.71rd 565 | . . . . 5 ⊢ (𝜑 → (𝑠 ∈ (𝑁‘𝑋) ↔ (𝑠 ∈ 𝒫 𝐵 ∧ 𝑠 ∈ (𝑁‘𝑋)))) |
17 | 16 | exbidv 1922 | . . . 4 ⊢ (𝜑 → (∃𝑠 𝑠 ∈ (𝑁‘𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ 𝑠 ∈ (𝑁‘𝑋)))) |
18 | 17 | bicomd 225 | . . 3 ⊢ (𝜑 → (∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ 𝑠 ∈ (𝑁‘𝑋)) ↔ ∃𝑠 𝑠 ∈ (𝑁‘𝑋))) |
19 | df-rex 3146 | . . 3 ⊢ (∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁‘𝑋) ↔ ∃𝑠(𝑠 ∈ 𝒫 𝐵 ∧ 𝑠 ∈ (𝑁‘𝑋))) | |
20 | n0 4312 | . . 3 ⊢ ((𝑁‘𝑋) ≠ ∅ ↔ ∃𝑠 𝑠 ∈ (𝑁‘𝑋)) | |
21 | 18, 19, 20 | 3bitr4g 316 | . 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑠 ∈ (𝑁‘𝑋) ↔ (𝑁‘𝑋) ≠ ∅)) |
22 | 9, 21 | bitrd 281 | 1 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ (𝑁‘𝑋) ≠ ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3018 ∃wrex 3141 {crab 3144 Vcvv 3496 ∅c0 4293 𝒫 cpw 4541 class class class wbr 5068 ↦ cmpt 5148 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ∈ cmpo 7160 ↑m cmap 8408 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 |
This theorem is referenced by: ntrneineine0 40444 |
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