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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 42288 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
5 | 1, 2, 4 | dssmapnvod 42274 | . 2 ⊢ (𝜑 → ◡𝐷 = 𝐷) |
6 | 1, 2, 3 | ntrclsf1o 42305 | . . . 4 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵)) |
7 | f1orel 6787 | . . . 4 ⊢ (𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵) → Rel 𝐷) | |
8 | relbrcnvg 6057 | . . . 4 ⊢ (Rel 𝐷 → (𝐾◡𝐷𝐼 ↔ 𝐼𝐷𝐾)) | |
9 | 6, 7, 8 | 3syl 18 | . . 3 ⊢ (𝜑 → (𝐾◡𝐷𝐼 ↔ 𝐼𝐷𝐾)) |
10 | 3, 9 | mpbird 256 | . 2 ⊢ (𝜑 → 𝐾◡𝐷𝐼) |
11 | 5, 10 | breqdi 5120 | 1 ⊢ (𝜑 → 𝐾𝐷𝐼) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 Vcvv 3445 ∖ cdif 3907 𝒫 cpw 4560 class class class wbr 5105 ↦ cmpt 5188 ◡ccnv 5632 Rel wrel 5638 –1-1-onto→wf1o 6495 ‘cfv 6496 (class class class)co 7356 ↑m cmap 8764 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7920 df-2nd 7921 df-map 8766 |
This theorem is referenced by: ntrclskex 42308 ntrclsfv2 42310 ntrclselnel2 42312 ntrclsfveq2 42315 ntrclsk4 42326 |
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