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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclsfveq1 | Structured version Visualization version GIF version |
Description: If interior and closure functions are related then specific function values are complementary. (Contributed by RP, 27-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) |
ntrclsfv.c | ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵) |
Ref | Expression |
---|---|
ntrclsfveq1 | ⊢ (𝜑 → ((𝐼‘𝑆) = 𝐶 ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrclsfv.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵) | |
2 | 1 | elpwid 4553 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) |
3 | dfss4 4235 | . . . . 5 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐶)) = 𝐶) | |
4 | 2, 3 | sylib 220 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ 𝐶)) = 𝐶) |
5 | 4 | eqcomd 2827 | . . 3 ⊢ (𝜑 → 𝐶 = (𝐵 ∖ (𝐵 ∖ 𝐶))) |
6 | 5 | eqeq2d 2832 | . 2 ⊢ (𝜑 → ((𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) = 𝐶 ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ (𝐵 ∖ 𝐶)))) |
7 | ntrcls.o | . . . 4 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) | |
8 | ntrcls.d | . . . 4 ⊢ 𝐷 = (𝑂‘𝐵) | |
9 | ntrcls.r | . . . 4 ⊢ (𝜑 → 𝐼𝐷𝐾) | |
10 | ntrclsfv.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
11 | 7, 8, 9, 10 | ntrclsfv 40402 | . . 3 ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))) |
12 | 11 | eqeq1d 2823 | . 2 ⊢ (𝜑 → ((𝐼‘𝑆) = 𝐶 ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) = 𝐶)) |
13 | 7, 8, 9 | ntrclskex 40397 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
14 | elmapi 8422 | . . . . . 6 ⊢ (𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵) | |
15 | 13, 14 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐾:𝒫 𝐵⟶𝒫 𝐵) |
16 | 8, 9 | ntrclsrcomplex 40378 | . . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
17 | 15, 16 | ffvelrnd 6847 | . . . 4 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑆)) ∈ 𝒫 𝐵) |
18 | 17 | elpwid 4553 | . . 3 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵) |
19 | difssd 4109 | . . 3 ⊢ (𝜑 → (𝐵 ∖ 𝐶) ⊆ 𝐵) | |
20 | rcompleq 40363 | . . 3 ⊢ (((𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵 ∧ (𝐵 ∖ 𝐶) ⊆ 𝐵) → ((𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ 𝐶) ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ (𝐵 ∖ 𝐶)))) | |
21 | 18, 19, 20 | syl2anc 586 | . 2 ⊢ (𝜑 → ((𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ 𝐶) ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ (𝐵 ∖ 𝐶)))) |
22 | 6, 12, 21 | 3bitr4d 313 | 1 ⊢ (𝜑 → ((𝐼‘𝑆) = 𝐶 ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 Vcvv 3495 ∖ cdif 3933 ⊆ wss 3936 𝒫 cpw 4539 class class class wbr 5059 ↦ cmpt 5139 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ↑m cmap 8400 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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 3497 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 4562 df-pr 4564 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-map 8402 |
This theorem is referenced by: ntrclsfveq 40405 |
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