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Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrclsfveq | Structured version Visualization version GIF version |
Description: If interior and closure functions are related then equality of a pair of function values is equivalent to equality of a pair of the other function's values. (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.t | ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵) |
Ref | Expression |
---|---|
ntrclsfveq | ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇)))) |
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 | ntrclsfv.t | . . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝒫 𝐵) | |
5 | 1, 2, 3, 4 | ntrclsfv 42743 | . . 3 ⊢ (𝜑 → (𝐼‘𝑇) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) |
6 | 5 | eqeq2d 2744 | . 2 ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))))) |
7 | ntrclsfv.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
8 | 2, 3 | ntrclsrcomplex 42719 | . . 3 ⊢ (𝜑 → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))) ∈ 𝒫 𝐵) |
9 | 1, 2, 3, 7, 8 | ntrclsfveq1 42744 | . 2 ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))))) |
10 | 1, 2, 3 | ntrclskex 42738 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
11 | elmapi 8839 | . . . . . . 7 ⊢ (𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵) | |
12 | 10, 11 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝐾:𝒫 𝐵⟶𝒫 𝐵) |
13 | 2, 3 | ntrclsrcomplex 42719 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∖ 𝑇) ∈ 𝒫 𝐵) |
14 | 12, 13 | ffvelcdmd 7083 | . . . . 5 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑇)) ∈ 𝒫 𝐵) |
15 | 14 | elpwid 4610 | . . . 4 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵) |
16 | dfss4 4257 | . . . 4 ⊢ ((𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) = (𝐾‘(𝐵 ∖ 𝑇))) | |
17 | 15, 16 | sylib 217 | . . 3 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) = (𝐾‘(𝐵 ∖ 𝑇))) |
18 | 17 | eqeq2d 2744 | . 2 ⊢ (𝜑 → ((𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇)))) |
19 | 6, 9, 18 | 3bitrd 305 | 1 ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∖ cdif 3944 ⊆ wss 3947 𝒫 cpw 4601 class class class wbr 5147 ↦ cmpt 5230 ⟶wf 6536 ‘cfv 6540 (class class class)co 7404 ↑m cmap 8816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-ov 7407 df-oprab 7408 df-mpo 7409 df-1st 7970 df-2nd 7971 df-map 8818 |
This theorem is referenced by: (None) |
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