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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclsfv1 | Structured version Visualization version GIF version |
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is a functional relation between them (Contributed by RP, 28-May-2021.) |
Ref | Expression |
---|---|
ntrcls.o | ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) |
ntrcls.d | ⊢ 𝐷 = (𝑂‘𝐵) |
ntrcls.r | ⊢ (𝜑 → 𝐼𝐷𝐾) |
Ref | Expression |
---|---|
ntrclsfv1 | ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrcls.r | . 2 ⊢ (𝜑 → 𝐼𝐷𝐾) | |
2 | ntrcls.o | . . . . . . 7 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) | |
3 | ntrcls.d | . . . . . . 7 ⊢ 𝐷 = (𝑂‘𝐵) | |
4 | 2, 3, 1 | ntrclsf1o 40754 | . . . . . 6 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵)) |
5 | f1ofn 6591 | . . . . . 6 ⊢ (𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵)) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵)) |
7 | 2, 3, 1 | ntrclsiex 40756 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
8 | 6, 7 | jca 515 | . . . 4 ⊢ (𝜑 → (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))) |
9 | fnfun 6423 | . . . . . 6 ⊢ (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → Fun 𝐷) | |
10 | 9 | adantr 484 | . . . . 5 ⊢ ((𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → Fun 𝐷) |
11 | fndm 6425 | . . . . . . 7 ⊢ (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → dom 𝐷 = (𝒫 𝐵 ↑m 𝒫 𝐵)) | |
12 | 11 | eleq2d 2875 | . . . . . 6 ⊢ (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → (𝐼 ∈ dom 𝐷 ↔ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))) |
13 | 12 | biimpar 481 | . . . . 5 ⊢ ((𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → 𝐼 ∈ dom 𝐷) |
14 | 10, 13 | jca 515 | . . . 4 ⊢ ((𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → (Fun 𝐷 ∧ 𝐼 ∈ dom 𝐷)) |
15 | 8, 14 | syl 17 | . . 3 ⊢ (𝜑 → (Fun 𝐷 ∧ 𝐼 ∈ dom 𝐷)) |
16 | funbrfvb 6695 | . . 3 ⊢ ((Fun 𝐷 ∧ 𝐼 ∈ dom 𝐷) → ((𝐷‘𝐼) = 𝐾 ↔ 𝐼𝐷𝐾)) | |
17 | 15, 16 | syl 17 | . 2 ⊢ (𝜑 → ((𝐷‘𝐼) = 𝐾 ↔ 𝐼𝐷𝐾)) |
18 | 1, 17 | mpbird 260 | 1 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 𝒫 cpw 4497 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 Fun wfun 6318 Fn wfn 6319 –1-1-onto→wf1o 6323 ‘cfv 6324 (class class class)co 7135 ↑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: ntrclsfv2 40759 ntrclscls00 40769 ntrclsiso 40770 ntrclsk2 40771 ntrclskb 40772 ntrclsk3 40773 ntrclsk13 40774 |
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