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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclselnel2 | Structured version Visualization version GIF version |
Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then there is an equivalence between membership in interior of the complement of a set and non-membership in the closure of the set. (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) |
ntrcls.x | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ntrcls.s | ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) |
Ref | Expression |
---|---|
ntrclselnel2 | ⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑆)) ↔ ¬ 𝑋 ∈ (𝐾‘𝑆))) |
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 | 1, 2, 3 | ntrclsnvobr 40422 | . . 3 ⊢ (𝜑 → 𝐾𝐷𝐼) |
5 | ntrcls.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
6 | ntrcls.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
7 | 1, 2, 4, 5, 6 | ntrclselnel1 40427 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝑆) ↔ ¬ 𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑆)))) |
8 | 7 | con2bid 357 | 1 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑆)) ↔ ¬ 𝑋 ∈ (𝐾‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∖ cdif 3933 𝒫 cpw 4539 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ↑m cmap 8406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-map 8408 |
This theorem is referenced by: ntrclsneine0lem 40434 |
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