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Theorem ntrneifv4 43789
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 3954 . 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 43780 . . . . . 6 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
6 elmapi 8870 . . . . . 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 7091 . . . 4 (𝜑 → (𝐼𝑆) ∈ 𝒫 𝐵)
109elpwid 4606 . . 3 (𝜑 → (𝐼𝑆) ⊆ 𝐵)
11 sseqin2 4213 . . 3 ((𝐼𝑆) ⊆ 𝐵 ↔ (𝐵 ∩ (𝐼𝑆)) = (𝐼𝑆))
1210, 11sylib 217 . 2 (𝜑 → (𝐵 ∩ (𝐼𝑆)) = (𝐼𝑆))
134adantr 479 . . . 4 ((𝜑𝑥𝐵) → 𝐼𝐹𝑁)
14 simpr 483 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐵)
158adantr 479 . . . 4 ((𝜑𝑥𝐵) → 𝑆 ∈ 𝒫 𝐵)
162, 3, 13, 14, 15ntrneiel 43785 . . 3 ((𝜑𝑥𝐵) → (𝑥 ∈ (𝐼𝑆) ↔ 𝑆 ∈ (𝑁𝑥)))
1716rabbidva 3426 . 2 (𝜑 → {𝑥𝐵𝑥 ∈ (𝐼𝑆)} = {𝑥𝐵𝑆 ∈ (𝑁𝑥)})
181, 12, 173eqtr3a 2790 1 (𝜑 → (𝐼𝑆) = {𝑥𝐵𝑆 ∈ (𝑁𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  {crab 3419  Vcvv 3462  cin 3945  wss 3946  𝒫 cpw 4597   class class class wbr 5145  cmpt 5228  wf 6542  cfv 6546  (class class class)co 7416  cmpo 7418  m cmap 8847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7995  df-2nd 7996  df-map 8849
This theorem is referenced by:  ntrneiel2  43790
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