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Theorem ntrneifv4 39153
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 3775 . 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 39144 . . . . . 6 (𝜑𝐼 ∈ (𝒫 𝐵𝑚 𝒫 𝐵))
6 elmapi 8115 . . . . . 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 6584 . . . 4 (𝜑 → (𝐼𝑆) ∈ 𝒫 𝐵)
109elpwid 4359 . . 3 (𝜑 → (𝐼𝑆) ⊆ 𝐵)
11 sseqin2 4013 . . 3 ((𝐼𝑆) ⊆ 𝐵 ↔ (𝐵 ∩ (𝐼𝑆)) = (𝐼𝑆))
1210, 11sylib 210 . 2 (𝜑 → (𝐵 ∩ (𝐼𝑆)) = (𝐼𝑆))
134adantr 473 . . . 4 ((𝜑𝑥𝐵) → 𝐼𝐹𝑁)
14 simpr 478 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐵)
158adantr 473 . . . 4 ((𝜑𝑥𝐵) → 𝑆 ∈ 𝒫 𝐵)
162, 3, 13, 14, 15ntrneiel 39149 . . 3 ((𝜑𝑥𝐵) → (𝑥 ∈ (𝐼𝑆) ↔ 𝑆 ∈ (𝑁𝑥)))
1716rabbidva 3370 . 2 (𝜑 → {𝑥𝐵𝑥 ∈ (𝐼𝑆)} = {𝑥𝐵𝑆 ∈ (𝑁𝑥)})
181, 12, 173eqtr3a 2855 1 (𝜑 → (𝐼𝑆) = {𝑥𝐵𝑆 ∈ (𝑁𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  {crab 3091  Vcvv 3383  cin 3766  wss 3767  𝒫 cpw 4347   class class class wbr 4841  cmpt 4920  wf 6095  cfv 6099  (class class class)co 6876  cmpt2 6878  𝑚 cmap 8093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-rep 4962  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-ral 3092  df-rex 3093  df-reu 3094  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-1st 7399  df-2nd 7400  df-map 8095
This theorem is referenced by:  ntrneiel2  39154
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