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Theorem ntrneifv4 41161
 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 3866 . 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 41152 . . . . . 6 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
6 elmapi 8438 . . . . . 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, 8ffvelrnd 6843 . . . 4 (𝜑 → (𝐼𝑆) ∈ 𝒫 𝐵)
109elpwid 4505 . . 3 (𝜑 → (𝐼𝑆) ⊆ 𝐵)
11 sseqin2 4120 . . 3 ((𝐼𝑆) ⊆ 𝐵 ↔ (𝐵 ∩ (𝐼𝑆)) = (𝐼𝑆))
1210, 11sylib 221 . 2 (𝜑 → (𝐵 ∩ (𝐼𝑆)) = (𝐼𝑆))
134adantr 484 . . . 4 ((𝜑𝑥𝐵) → 𝐼𝐹𝑁)
14 simpr 488 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐵)
158adantr 484 . . . 4 ((𝜑𝑥𝐵) → 𝑆 ∈ 𝒫 𝐵)
162, 3, 13, 14, 15ntrneiel 41157 . . 3 ((𝜑𝑥𝐵) → (𝑥 ∈ (𝐼𝑆) ↔ 𝑆 ∈ (𝑁𝑥)))
1716rabbidva 3390 . 2 (𝜑 → {𝑥𝐵𝑥 ∈ (𝐼𝑆)} = {𝑥𝐵𝑆 ∈ (𝑁𝑥)})
181, 12, 173eqtr3a 2817 1 (𝜑 → (𝐼𝑆) = {𝑥𝐵𝑆 ∈ (𝑁𝑥)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  {crab 3074  Vcvv 3409   ∩ cin 3857   ⊆ wss 3858  𝒫 cpw 4494   class class class wbr 5032   ↦ cmpt 5112  ⟶wf 6331  ‘cfv 6335  (class class class)co 7150   ∈ cmpo 7152   ↑m cmap 8416 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7693  df-2nd 7694  df-map 8418 This theorem is referenced by:  ntrneiel2  41162
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