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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 40778 | . . . 4 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵 ↑m 𝐵)) |
5 | f1orel 6593 | . . . 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 5778 | . . 3 ⊢ ((Rel 𝐹 ∧ 𝐼𝐹𝑁) → 𝐼 ∈ dom 𝐹) | |
8 | 6, 3, 7 | syl2anc 587 | . 2 ⊢ (𝜑 → 𝐼 ∈ dom 𝐹) |
9 | f1odm 6594 | . . 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 2892 | 1 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {crab 3110 Vcvv 3441 𝒫 cpw 4497 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 Rel wrel 5524 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 ∈ cmpo 7137 ↑m cmap 8389 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 |
This theorem is referenced by: ntrneifv1 40782 ntrneifv2 40783 ntrneiel 40784 ntrneifv4 40788 ntrneiel2 40789 ntrneicls00 40792 ntrneicls11 40793 ntrneiiso 40794 ntrneik2 40795 ntrneikb 40797 ntrneixb 40798 ntrneik3 40799 ntrneix3 40800 ntrneik13 40801 ntrneix13 40802 ntrneik4w 40803 ntrneik4 40804 |
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