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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclsiex | Structured version Visualization version GIF version | ||
| Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then those functions are maps of subsets to subsets. (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 |
|---|---|
| ntrclsiex | ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ntrcls.o | . . . . 5 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) | |
| 2 | ntrcls.d | . . . . 5 ⊢ 𝐷 = (𝑂‘𝐵) | |
| 3 | ntrcls.r | . . . . 5 ⊢ (𝜑 → 𝐼𝐷𝐾) | |
| 4 | 1, 2, 3 | ntrclsf1o 44064 | . . . 4 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵)) |
| 5 | f1orel 6851 | . . . 4 ⊢ (𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵) → Rel 𝐷) | |
| 6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → Rel 𝐷) |
| 7 | releldm 5955 | . . 3 ⊢ ((Rel 𝐷 ∧ 𝐼𝐷𝐾) → 𝐼 ∈ dom 𝐷) | |
| 8 | 6, 3, 7 | syl2anc 584 | . 2 ⊢ (𝜑 → 𝐼 ∈ dom 𝐷) |
| 9 | f1odm 6852 | . . 3 ⊢ (𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵) → dom 𝐷 = (𝒫 𝐵 ↑m 𝒫 𝐵)) | |
| 10 | 4, 9 | syl 17 | . 2 ⊢ (𝜑 → dom 𝐷 = (𝒫 𝐵 ↑m 𝒫 𝐵)) |
| 11 | 8, 10 | eleqtrd 2843 | 1 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 𝒫 cpw 4600 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 Rel wrel 5690 –1-1-onto→wf1o 6560 ‘cfv 6561 (class class class)co 7431 ↑m cmap 8866 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 |
| This theorem is referenced by: ntrclskex 44067 ntrclsfv1 44068 ntrclsfveq2 44074 ntrclscls00 44079 ntrclsiso 44080 ntrclsk2 44081 ntrclskb 44082 ntrclsk3 44083 ntrclsk13 44084 ntrclsk4 44085 clsneikex 44119 |
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