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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 40287 | . . 3 ⊢ (𝜑 → (𝐼‘𝑇) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) |
6 | 5 | eqeq2d 2829 | . 2 ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))))) |
7 | ntrclsfv.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
8 | 2, 3 | ntrclsrcomplex 40263 | . . 3 ⊢ (𝜑 → (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))) ∈ 𝒫 𝐵) |
9 | 1, 2, 3, 7, 8 | ntrclsfveq1 40288 | . 2 ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇))) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))))) |
10 | 1, 2, 3 | ntrclskex 40282 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
11 | elmapi 8417 | . . . . . . 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 40263 | . . . . . 6 ⊢ (𝜑 → (𝐵 ∖ 𝑇) ∈ 𝒫 𝐵) |
14 | 12, 13 | ffvelrnd 6844 | . . . . 5 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑇)) ∈ 𝒫 𝐵) |
15 | 14 | elpwid 4549 | . . . 4 ⊢ (𝜑 → (𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵) |
16 | dfss4 4232 | . . . 4 ⊢ ((𝐾‘(𝐵 ∖ 𝑇)) ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) = (𝐾‘(𝐵 ∖ 𝑇))) | |
17 | 15, 16 | sylib 219 | . . 3 ⊢ (𝜑 → (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) = (𝐾‘(𝐵 ∖ 𝑇))) |
18 | 17 | eqeq2d 2829 | . 2 ⊢ (𝜑 → ((𝐾‘(𝐵 ∖ 𝑆)) = (𝐵 ∖ (𝐵 ∖ (𝐾‘(𝐵 ∖ 𝑇)))) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇)))) |
19 | 6, 9, 18 | 3bitrd 306 | 1 ⊢ (𝜑 → ((𝐼‘𝑆) = (𝐼‘𝑇) ↔ (𝐾‘(𝐵 ∖ 𝑆)) = (𝐾‘(𝐵 ∖ 𝑇)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 Vcvv 3492 ∖ cdif 3930 ⊆ wss 3933 𝒫 cpw 4535 class class class wbr 5057 ↦ cmpt 5137 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-map 8397 |
This theorem is referenced by: (None) |
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