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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclsfveq2 | 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 |
---|---|
ntrclsfveq2 | ⊢ (𝜑 → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐾‘𝑆) = (𝐵 ∖ 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ntrcls.o | . . . . . . 7 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑m 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗)))))) | |
2 | ntrcls.d | . . . . . . 7 ⊢ 𝐷 = (𝑂‘𝐵) | |
3 | ntrcls.r | . . . . . . 7 ⊢ (𝜑 → 𝐼𝐷𝐾) | |
4 | 1, 2, 3 | ntrclsiex 40401 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
5 | elmapi 8427 | . . . . . 6 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵) |
7 | 2, 3 | ntrclsrcomplex 40383 | . . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
8 | 6, 7 | ffvelrnd 6851 | . . . 4 ⊢ (𝜑 → (𝐼‘(𝐵 ∖ 𝑆)) ∈ 𝒫 𝐵) |
9 | 8 | elpwid 4549 | . . 3 ⊢ (𝜑 → (𝐼‘(𝐵 ∖ 𝑆)) ⊆ 𝐵) |
10 | ntrclsfv.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵) | |
11 | 10 | elpwid 4549 | . . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) |
12 | rcompleq 40368 | . . 3 ⊢ (((𝐼‘(𝐵 ∖ 𝑆)) ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵) → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ 𝐶))) | |
13 | 9, 11, 12 | syl2anc 586 | . 2 ⊢ (𝜑 → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ 𝐶))) |
14 | 1, 2, 3 | ntrclsnvobr 40400 | . . . 4 ⊢ (𝜑 → 𝐾𝐷𝐼) |
15 | ntrclsfv.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
16 | 1, 2, 14, 15 | ntrclsfv 40407 | . . 3 ⊢ (𝜑 → (𝐾‘𝑆) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆)))) |
17 | 16 | eqeq1d 2823 | . 2 ⊢ (𝜑 → ((𝐾‘𝑆) = (𝐵 ∖ 𝐶) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ 𝐶))) |
18 | 13, 17 | bitr4d 284 | 1 ⊢ (𝜑 → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐾‘𝑆) = (𝐵 ∖ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∖ cdif 3932 ⊆ wss 3935 𝒫 cpw 4538 class class class wbr 5065 ↦ cmpt 5145 ⟶wf 6350 ‘cfv 6354 (class class class)co 7155 ↑m cmap 8405 |
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 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
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 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-map 8407 |
This theorem is referenced by: (None) |
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