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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrneiiex | Structured version Visualization version GIF version |
Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the interior function exists. (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) |
Ref | Expression |
---|---|
ntrneiiex | ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrnei.o | . . . . 5 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
2 | ntrnei.f | . . . . 5 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
3 | ntrnei.r | . . . . 5 ⊢ (𝜑 → 𝐼𝐹𝑁) | |
4 | 1, 2, 3 | ntrneif1o 44065 | . . . 4 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) |
5 | f1orel 6852 | . . . 4 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → Rel 𝐹) | |
6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → Rel 𝐹) |
7 | releldm 5958 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐼𝐹𝑁) → 𝐼 ∈ dom 𝐹) | |
8 | 6, 3, 7 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐼 ∈ dom 𝐹) |
9 | f1odm 6853 | . . 3 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵) → dom 𝐹 = (𝒫 𝐵 ↑m 𝒫 𝐵)) | |
10 | 4, 9 | syl 17 | . 2 ⊢ (𝜑 → dom 𝐹 = (𝒫 𝐵 ↑m 𝒫 𝐵)) |
11 | 8, 10 | eleqtrd 2841 | 1 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 𝒫 cpw 4605 class class class wbr 5148 ↦ cmpt 5231 dom cdm 5689 Rel wrel 5694 –1-1-onto→wf1o 6562 ‘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: ntrneifv1 44069 ntrneifv2 44070 ntrneiel 44071 ntrneifv4 44075 ntrneiel2 44076 ntrneicls00 44079 ntrneicls11 44080 ntrneiiso 44081 ntrneik2 44082 ntrneikb 44084 ntrneixb 44085 ntrneik3 44086 ntrneix3 44087 ntrneik13 44088 ntrneix13 44089 ntrneik4w 44090 ntrneik4 44091 |
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