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Theorem ntrneifv4 40319
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 3948 . 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 40310 . . . . . 6 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
6 elmapi 8423 . . . . . 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 6850 . . . 4 (𝜑 → (𝐼𝑆) ∈ 𝒫 𝐵)
109elpwid 4556 . . 3 (𝜑 → (𝐼𝑆) ⊆ 𝐵)
11 sseqin2 4196 . . 3 ((𝐼𝑆) ⊆ 𝐵 ↔ (𝐵 ∩ (𝐼𝑆)) = (𝐼𝑆))
1210, 11sylib 219 . 2 (𝜑 → (𝐵 ∩ (𝐼𝑆)) = (𝐼𝑆))
134adantr 481 . . . 4 ((𝜑𝑥𝐵) → 𝐼𝐹𝑁)
14 simpr 485 . . . 4 ((𝜑𝑥𝐵) → 𝑥𝐵)
158adantr 481 . . . 4 ((𝜑𝑥𝐵) → 𝑆 ∈ 𝒫 𝐵)
162, 3, 13, 14, 15ntrneiel 40315 . . 3 ((𝜑𝑥𝐵) → (𝑥 ∈ (𝐼𝑆) ↔ 𝑆 ∈ (𝑁𝑥)))
1716rabbidva 3484 . 2 (𝜑 → {𝑥𝐵𝑥 ∈ (𝐼𝑆)} = {𝑥𝐵𝑆 ∈ (𝑁𝑥)})
181, 12, 173eqtr3a 2885 1 (𝜑 → (𝐼𝑆) = {𝑥𝐵𝑆 ∈ (𝑁𝑥)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  {crab 3147  Vcvv 3500  cin 3939  wss 3940  𝒫 cpw 4542   class class class wbr 5063  cmpt 5143  wf 6350  cfv 6354  (class class class)co 7150  cmpo 7152  m cmap 8401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-map 8403
This theorem is referenced by:  ntrneiel2  40320
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