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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 42330 | . . . . . 6 ⊢ (𝜑 → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵)) |
5 | f1ofn 6786 | . . . . . 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 42332 | . . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
8 | 6, 7 | jca 513 | . . . 4 ⊢ (𝜑 → (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))) |
9 | fnfun 6603 | . . . . . 6 ⊢ (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → Fun 𝐷) | |
10 | 9 | adantr 482 | . . . . 5 ⊢ ((𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → Fun 𝐷) |
11 | fndm 6606 | . . . . . . 7 ⊢ (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → dom 𝐷 = (𝒫 𝐵 ↑m 𝒫 𝐵)) | |
12 | 11 | eleq2d 2824 | . . . . . 6 ⊢ (𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) → (𝐼 ∈ dom 𝐷 ↔ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))) |
13 | 12 | biimpar 479 | . . . . 5 ⊢ ((𝐷 Fn (𝒫 𝐵 ↑m 𝒫 𝐵) ∧ 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) → 𝐼 ∈ dom 𝐷) |
14 | 10, 13 | jca 513 | . . . 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 6898 | . . 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 257 | 1 ⊢ (𝜑 → (𝐷‘𝐼) = 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3446 ∖ cdif 3908 𝒫 cpw 4561 class class class wbr 5106 ↦ cmpt 5189 dom cdm 5634 Fun wfun 6491 Fn wfn 6492 –1-1-onto→wf1o 6496 ‘cfv 6497 (class class class)co 7358 ↑m cmap 8766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3355 df-rab 3409 df-v 3448 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-ov 7361 df-oprab 7362 df-mpo 7363 df-1st 7922 df-2nd 7923 df-map 8768 |
This theorem is referenced by: ntrclsfv2 42335 ntrclscls00 42345 ntrclsiso 42346 ntrclsk2 42347 ntrclskb 42348 ntrclsk3 42349 ntrclsk13 42350 |
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