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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclsnvobr | Structured version Visualization version GIF version |
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then they are related the opposite way. (Contributed by RP, 21-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 |
---|---|
ntrclsnvobr | ⊢ (𝜑 → 𝐾𝐷𝐼) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrcls.o | . . 3 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) | |
2 | ntrcls.d | . . 3 ⊢ 𝐷 = (𝑂‘𝐵) | |
3 | ntrcls.r | . . . 4 ⊢ (𝜑 → 𝐼𝐷𝐾) | |
4 | 2, 3 | ntrclsbex 38332 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
5 | 1, 2, 4 | dssmapnvod 38314 | . 2 ⊢ (𝜑 → ◡𝐷 = 𝐷) |
6 | 1, 2, 3 | ntrclsf1o 38349 | . . . 4 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
7 | f1orel 6140 | . . . 4 ⊢ (𝐷:(𝒫 𝐵 ↑𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑𝑚 𝒫 𝐵) → Rel 𝐷) | |
8 | relbrcnvg 5504 | . . . 4 ⊢ (Rel 𝐷 → (𝐾◡𝐷𝐼 ↔ 𝐼𝐷𝐾)) | |
9 | 6, 7, 8 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐾◡𝐷𝐼 ↔ 𝐼𝐷𝐾)) |
10 | 3, 9 | mpbird 247 | . 2 ⊢ (𝜑 → 𝐾◡𝐷𝐼) |
11 | 5, 10 | breqdi 4668 | 1 ⊢ (𝜑 → 𝐾𝐷𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1483 Vcvv 3200 ∖ cdif 3571 𝒫 cpw 4158 class class class wbr 4653 ↦ cmpt 4729 ◡ccnv 5113 Rel wrel 5119 –1-1-onto→wf1o 5887 ‘cfv 5888 (class class class)co 6650 ↑𝑚 cmap 7857 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-map 7859 |
This theorem is referenced by: ntrclskex 38352 ntrclsfv2 38354 ntrclselnel2 38356 ntrclsfveq2 38359 ntrclsk4 38370 |
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