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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ntrneifv4 | Structured version Visualization version GIF version | ||
| Description: The value of the interior (closure) expressed in terms of the neighbors (convergents) function. (Contributed by RP, 26-Jun-2021.) |
| Ref | Expression |
|---|---|
| ntrnei.o | ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) |
| ntrnei.f | ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) |
| ntrnei.r | ⊢ (𝜑 → 𝐼𝐹𝑁) |
| ntrneifv.s | ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) |
| Ref | Expression |
|---|---|
| ntrneifv4 | ⊢ (𝜑 → (𝐼‘𝑆) = {𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfin5 3959 | . 2 ⊢ (𝐵 ∩ (𝐼‘𝑆)) = {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ (𝐼‘𝑆)} | |
| 2 | ntrnei.o | . . . . . . 7 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)}))) | |
| 3 | ntrnei.f | . . . . . . 7 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵) | |
| 4 | ntrnei.r | . . . . . . 7 ⊢ (𝜑 → 𝐼𝐹𝑁) | |
| 5 | 2, 3, 4 | ntrneiiex 44089 | . . . . . 6 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵)) |
| 6 | elmapi 8889 | . . . . . 6 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵) | |
| 7 | 5, 6 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵) |
| 8 | ntrneifv.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵) | |
| 9 | 7, 8 | ffvelcdmd 7105 | . . . 4 ⊢ (𝜑 → (𝐼‘𝑆) ∈ 𝒫 𝐵) |
| 10 | 9 | elpwid 4609 | . . 3 ⊢ (𝜑 → (𝐼‘𝑆) ⊆ 𝐵) |
| 11 | sseqin2 4223 | . . 3 ⊢ ((𝐼‘𝑆) ⊆ 𝐵 ↔ (𝐵 ∩ (𝐼‘𝑆)) = (𝐼‘𝑆)) | |
| 12 | 10, 11 | sylib 218 | . 2 ⊢ (𝜑 → (𝐵 ∩ (𝐼‘𝑆)) = (𝐼‘𝑆)) |
| 13 | 4 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐼𝐹𝑁) |
| 14 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
| 15 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑆 ∈ 𝒫 𝐵) |
| 16 | 2, 3, 13, 14, 15 | ntrneiel 44094 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑆) ↔ 𝑆 ∈ (𝑁‘𝑥))) |
| 17 | 16 | rabbidva 3443 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ (𝐼‘𝑆)} = {𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)}) |
| 18 | 1, 12, 17 | 3eqtr3a 2801 | 1 ⊢ (𝜑 → (𝐼‘𝑆) = {𝑥 ∈ 𝐵 ∣ 𝑆 ∈ (𝑁‘𝑥)}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {crab 3436 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 class class class wbr 5143 ↦ cmpt 5225 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 ∈ cmpo 7433 ↑m cmap 8866 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-map 8868 |
| This theorem is referenced by: ntrneiel2 44099 |
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