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Theorem ntrneifv4 44098
Description: The value of the interior (closure) expressed in terms of the neighbors (convergents) function. (Contributed by RP, 26-Jun-2021.)
Hypotheses
Ref Expression
ntrnei.o 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
ntrnei.f 𝐹 = (𝒫 𝐵𝑂𝐵)
ntrnei.r (𝜑𝐼𝐹𝑁)
ntrneifv.s (𝜑𝑆 ∈ 𝒫 𝐵)
Assertion
Ref Expression
ntrneifv4 (𝜑 → (𝐼𝑆) = {𝑥𝐵𝑆 ∈ (𝑁𝑥)})
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑥   𝑘,𝐼,𝑙,𝑚,𝑥   𝑆,𝑚,𝑥   𝜑,𝑖,𝑗,𝑘,𝑙,𝑥
Allowed substitution hints:   𝜑(𝑚)   𝑆(𝑖,𝑗,𝑘,𝑙)   𝐹(𝑥,𝑖,𝑗,𝑘,𝑚,𝑙)   𝐼(𝑖,𝑗)   𝑁(𝑥,𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑥,𝑖,𝑗,𝑘,𝑚,𝑙)

Proof of Theorem ntrneifv4
StepHypRef Expression
1 dfin5 3959 . 2 (𝐵 ∩ (𝐼𝑆)) = {𝑥𝐵𝑥 ∈ (𝐼𝑆)}
2 ntrnei.o . . . . . . 7 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
3 ntrnei.f . . . . . . 7 𝐹 = (𝒫 𝐵𝑂𝐵)
4 ntrnei.r . . . . . . 7 (𝜑𝐼𝐹𝑁)
52, 3, 4ntrneiiex 44089 . . . . . 6 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
6 elmapi 8889 . . . . . 6 (𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
75, 6syl 17 . . . . 5 (𝜑𝐼:𝒫 𝐵⟶𝒫 𝐵)
8 ntrneifv.s . . . . 5 (𝜑𝑆 ∈ 𝒫 𝐵)
97, 8ffvelcdmd 7105 . . . 4 (𝜑 → (𝐼𝑆) ∈ 𝒫 𝐵)
109elpwid 4609 . . 3 (𝜑 → (𝐼𝑆) ⊆ 𝐵)
11 sseqin2 4223 . . 3 ((𝐼𝑆) ⊆ 𝐵 ↔ (𝐵 ∩ (𝐼𝑆)) = (𝐼𝑆))
1210, 11sylib 218 . 2 (𝜑 → (𝐵 ∩ (𝐼𝑆)) = (𝐼𝑆))
134adantr 480 . . . 4 ((𝜑𝑥𝐵) → 𝐼𝐹𝑁)
14 simpr 484 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐵)
158adantr 480 . . . 4 ((𝜑𝑥𝐵) → 𝑆 ∈ 𝒫 𝐵)
162, 3, 13, 14, 15ntrneiel 44094 . . 3 ((𝜑𝑥𝐵) → (𝑥 ∈ (𝐼𝑆) ↔ 𝑆 ∈ (𝑁𝑥)))
1716rabbidva 3443 . 2 (𝜑 → {𝑥𝐵𝑥 ∈ (𝐼𝑆)} = {𝑥𝐵𝑆 ∈ (𝑁𝑥)})
181, 12, 173eqtr3a 2801 1 (𝜑 → (𝐼𝑆) = {𝑥𝐵𝑆 ∈ (𝑁𝑥)})
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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