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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 4606 | . . . . 5 ⊢ (𝜑 → 𝐶 ⊆ 𝐵) |
3 | dfss4 4255 | . . . . 5 ⊢ (𝐶 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝐶)) = 𝐶) | |
4 | 2, 3 | sylib 217 | . . . 4 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ 𝐶)) = 𝐶) |
5 | 4 | eqcomd 2738 | . . 3 ⊢ (𝜑 → 𝐶 = (𝐵 ∖ (𝐵 ∖ 𝐶))) |
6 | 5 | eqeq2d 2743 | . 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 42645 | . . 3 ⊢ (𝜑 → (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆)))) |
12 | 11 | eqeq1d 2734 | . 2 ⊢ (𝜑 → ((𝐼‘𝑆) = 𝐶 ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) = 𝐶)) |
13 | 7, 8, 9 | ntrclskex 42640 | . . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
14 | elmapi 8828 | . . . . . 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 42621 | . . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵) |
17 | 15, 16 | ffvelcdmd 7073 | . . . 4 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑆)) ∈ 𝒫 𝐵) |
18 | 17 | elpwid 4606 | . . 3 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑆)) ⊆ 𝐵) |
19 | difssd 4129 | . . 3 ⊢ (𝜑 → (𝐵 ∖ 𝐶) ⊆ 𝐵) | |
20 | rcompleq 4292 | . . 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 584 | . 2 ⊢ (𝜑 → ((𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ 𝐶) ↔ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑆))) = (𝐵 ∖ (𝐵 ∖ 𝐶)))) |
22 | 6, 12, 21 | 3bitr4d 310 | 1 ⊢ (𝜑 → ((𝐼‘𝑆) = 𝐶 ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ 𝐶))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∖ cdif 3942 ⊆ wss 3945 𝒫 cpw 4597 class class class wbr 5142 ↦ cmpt 5225 ⟶wf 6529 ‘cfv 6533 (class class class)co 7394 ↑m cmap 8805 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7397 df-oprab 7398 df-mpo 7399 df-1st 7959 df-2nd 7960 df-map 8807 |
This theorem is referenced by: ntrclsfveq 42648 |
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