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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 44043 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
5 | elmapi 8888 | . . . . . 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 44025 | . . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
8 | 6, 7 | ffvelcdmd 7105 | . . . 4 ⊢ (𝜑 → (𝐼‘(𝐵 ∖ 𝑆)) ∈ 𝒫 𝐵) |
9 | 8 | elpwid 4614 | . . 3 ⊢ (𝜑 → (𝐼‘(𝐵 ∖ 𝑆)) ⊆ 𝐵) |
10 | ntrclsfv.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝒫 𝐵) | |
11 | 10 | elpwid 4614 | . . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) |
12 | rcompleq 4311 | . . 3 ⊢ (((𝐼‘(𝐵 ∖ 𝑆)) ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐵) → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ 𝐶))) | |
13 | 9, 11, 12 | syl2anc 584 | . 2 ⊢ (𝜑 → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ 𝐶))) |
14 | 1, 2, 3 | ntrclsnvobr 44042 | . . . 4 ⊢ (𝜑 → 𝐾𝐷𝐼) |
15 | ntrclsfv.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
16 | 1, 2, 14, 15 | ntrclsfv 44049 | . . 3 ⊢ (𝜑 → (𝐾‘𝑆) = (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆)))) |
17 | 16 | eqeq1d 2737 | . 2 ⊢ (𝜑 → ((𝐾‘𝑆) = (𝐵 ∖ 𝐶) ↔ (𝐵 ∖ (𝐼‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ 𝐶))) |
18 | 13, 17 | bitr4d 282 | 1 ⊢ (𝜑 → ((𝐼‘(𝐵 ∖ 𝑆)) = 𝐶 ↔ (𝐾‘𝑆) = (𝐵 ∖ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∖ cdif 3960 ⊆ wss 3963 𝒫 cpw 4605 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-map 8867 |
This theorem is referenced by: (None) |
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