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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclsfv | Structured version Visualization version GIF version |
Description: The value of the interior (closure) expressed in terms of the closure (interior). (Contributed by RP, 25-Jun-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) |
ntrclsfv.s | ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) |
Ref | Expression |
---|---|
ntrclsfv | ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrcls.o | . . . 4 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) | |
2 | ntrcls.d | . . . 4 ⊢ 𝐷 = (𝑂‘𝐵) | |
3 | ntrcls.r | . . . 4 ⊢ (𝜑 → 𝐼𝐷𝐾) | |
4 | 1, 2, 3 | ntrclsfv2 38878 | . . 3 ⊢ (𝜑 → (𝐷‘𝐾) = 𝐼) |
5 | 4 | fveq1d 6335 | . 2 ⊢ (𝜑 → ((𝐷‘𝐾)‘𝑆) = (𝐼‘𝑆)) |
6 | 2, 3 | ntrclsbex 38856 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
7 | 1, 2, 3 | ntrclskex 38876 | . . 3 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵)) |
8 | eqid 2771 | . . 3 ⊢ (𝐷‘𝐾) = (𝐷‘𝐾) | |
9 | ntrclsfv.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
10 | eqid 2771 | . . 3 ⊢ ((𝐷‘𝐾)‘𝑆) = ((𝐷‘𝐾)‘𝑆) | |
11 | 1, 2, 6, 7, 8, 9, 10 | dssmapfv3d 38837 | . 2 ⊢ (𝜑 → ((𝐷‘𝐾)‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))) |
12 | 5, 11 | eqtr3d 2807 | 1 ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 Vcvv 3351 ∖ cdif 3720 𝒫 cpw 4298 class class class wbr 4787 ↦ cmpt 4864 ‘cfv 6030 (class class class)co 6796 ↑𝑚 cmap 8013 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4905 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-iun 4657 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-res 5262 df-ima 5263 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-f1 6035 df-fo 6036 df-f1o 6037 df-fv 6038 df-ov 6799 df-oprab 6800 df-mpt2 6801 df-1st 7319 df-2nd 7320 df-map 8015 |
This theorem is referenced by: ntrclsfveq1 38882 ntrclsfveq2 38883 ntrclsfveq 38884 ntrclsss 38885 ntrclscls00 38888 ntrclsk4 38894 |
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