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Theorem ntrneinex 39157
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 39155 . . . 4 (𝜑𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))
5 f1orel 6359 . . . 4 (𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵) → Rel 𝐹)
64, 5syl 17 . . 3 (𝜑 → Rel 𝐹)
7 relelrn 5563 . . 3 ((Rel 𝐹𝐼𝐹𝑁) → 𝑁 ∈ ran 𝐹)
86, 3, 7syl2anc 580 . 2 (𝜑𝑁 ∈ ran 𝐹)
9 dff1o2 6361 . . . 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 210 . . 3 (𝜑 → (𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ Fun 𝐹 ∧ ran 𝐹 = (𝒫 𝒫 𝐵𝑚 𝐵)))
1110simp3d 1175 . 2 (𝜑 → ran 𝐹 = (𝒫 𝒫 𝐵𝑚 𝐵))
128, 11eleqtrd 2880 1 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1108   = wceq 1653  wcel 2157  {crab 3093  Vcvv 3385  𝒫 cpw 4349   class class class wbr 4843  cmpt 4922  ccnv 5311  ran crn 5313  Rel wrel 5317  Fun wfun 6095   Fn wfn 6096  1-1-ontowf1o 6100  cfv 6101  (class class class)co 6878  cmpt2 6880  𝑚 cmap 8095
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 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
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 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-map 8097
This theorem is referenced by:  ntrneifv2  39160  ntrneifv3  39162  ntrneineine0lem  39163  ntrneineine1lem  39164  ntrneiel2  39166  clsneinex  39187  neicvgmex  39197
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