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Theorem ntrneinex 41640
Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the neighborhood function exists. (Contributed by RP, 29-May-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)
Assertion
Ref Expression
ntrneinex (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚   𝜑,𝑖,𝑗,𝑘,𝑙
Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑖,𝑗,𝑘,𝑚,𝑙)   𝐼(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑁(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑙)

Proof of Theorem ntrneinex
StepHypRef Expression
1 ntrnei.o . . . . 5 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗m 𝑖) ↦ (𝑙𝑗 ↦ {𝑚𝑖𝑙 ∈ (𝑘𝑚)})))
2 ntrnei.f . . . . 5 𝐹 = (𝒫 𝐵𝑂𝐵)
3 ntrnei.r . . . . 5 (𝜑𝐼𝐹𝑁)
41, 2, 3ntrneif1o 41638 . . . 4 (𝜑𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
5 f1orel 6715 . . . 4 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) → Rel 𝐹)
64, 5syl 17 . . 3 (𝜑 → Rel 𝐹)
7 relelrn 5851 . . 3 ((Rel 𝐹𝐼𝐹𝑁) → 𝑁 ∈ ran 𝐹)
86, 3, 7syl2anc 583 . 2 (𝜑𝑁 ∈ ran 𝐹)
9 dff1o2 6717 . . . 4 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) ↔ (𝐹 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ Fun 𝐹 ∧ ran 𝐹 = (𝒫 𝒫 𝐵m 𝐵)))
104, 9sylib 217 . . 3 (𝜑 → (𝐹 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ Fun 𝐹 ∧ ran 𝐹 = (𝒫 𝒫 𝐵m 𝐵)))
1110simp3d 1142 . 2 (𝜑 → ran 𝐹 = (𝒫 𝒫 𝐵m 𝐵))
128, 11eleqtrd 2842 1 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085   = wceq 1541  wcel 2109  {crab 3069  Vcvv 3430  𝒫 cpw 4538   class class class wbr 5078  cmpt 5161  ccnv 5587  ran crn 5589  Rel wrel 5593  Fun wfun 6424   Fn wfn 6425  1-1-ontowf1o 6429  cfv 6430  (class class class)co 7268  cmpo 7270  m cmap 8589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-rep 5213  ax-sep 5226  ax-nul 5233  ax-pow 5291  ax-pr 5355  ax-un 7579
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-reu 3072  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-iun 4931  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-f1 6435  df-fo 6436  df-f1o 6437  df-fv 6438  df-ov 7271  df-oprab 7272  df-mpo 7273  df-1st 7817  df-2nd 7818  df-map 8591
This theorem is referenced by:  ntrneifv2  41643  ntrneifv3  41645  ntrneineine0lem  41646  ntrneineine1lem  41647  ntrneiel2  41649  clsneinex  41670  neicvgmex  41680
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