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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 41634 | . . 3 ⊢ (𝜑 → (𝐷‘𝐾) = 𝐼) |
5 | 4 | fveq1d 6771 | . 2 ⊢ (𝜑 → ((𝐷‘𝐾)‘𝑆) = (𝐼‘𝑆)) |
6 | 2, 3 | ntrclsbex 41612 | . . 3 ⊢ (𝜑 → 𝐵 ∈ V) |
7 | 1, 2, 3 | ntrclskex 41632 | . . 3 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
8 | eqid 2740 | . . 3 ⊢ (𝐷‘𝐾) = (𝐷‘𝐾) | |
9 | ntrclsfv.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
10 | eqid 2740 | . . 3 ⊢ ((𝐷‘𝐾)‘𝑆) = ((𝐷‘𝐾)‘𝑆) | |
11 | 1, 2, 6, 7, 8, 9, 10 | dssmapfv3d 41595 | . 2 ⊢ (𝜑 → ((𝐷‘𝐾)‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))) |
12 | 5, 11 | eqtr3d 2782 | 1 ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2110 Vcvv 3431 ∖ cdif 3889 𝒫 cpw 4539 class class class wbr 5079 ↦ cmpt 5162 ‘cfv 6431 (class class class)co 7269 ↑m cmap 8596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-ov 7272 df-oprab 7273 df-mpo 7274 df-1st 7822 df-2nd 7823 df-map 8598 |
This theorem is referenced by: ntrclsfveq1 41638 ntrclsfveq2 41639 ntrclsfveq 41640 ntrclsss 41641 ntrclscls00 41644 ntrclsk4 41650 |
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