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Theorem ntrneinex 40562
 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 40560 . . . 4 (𝜑𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
5 f1orel 6594 . . . 4 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) → Rel 𝐹)
64, 5syl 17 . . 3 (𝜑 → Rel 𝐹)
7 relelrn 5791 . . 3 ((Rel 𝐹𝐼𝐹𝑁) → 𝑁 ∈ ran 𝐹)
86, 3, 7syl2anc 586 . 2 (𝜑𝑁 ∈ ran 𝐹)
9 dff1o2 6596 . . . 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 220 . . 3 (𝜑 → (𝐹 Fn (𝒫 𝐵m 𝒫 𝐵) ∧ Fun 𝐹 ∧ ran 𝐹 = (𝒫 𝒫 𝐵m 𝐵)))
1110simp3d 1140 . 2 (𝜑 → ran 𝐹 = (𝒫 𝒫 𝐵m 𝐵))
128, 11eleqtrd 2913 1 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵m 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  {crab 3129  Vcvv 3473  𝒫 cpw 4515   class class class wbr 5042   ↦ cmpt 5122  ◡ccnv 5530  ran crn 5532  Rel wrel 5536  Fun wfun 6325   Fn wfn 6326  –1-1-onto→wf1o 6330  ‘cfv 6331  (class class class)co 7133   ∈ cmpo 7135   ↑m cmap 8384 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-1st 7667  df-2nd 7668  df-map 8386 This theorem is referenced by:  ntrneifv2  40565  ntrneifv3  40567  ntrneineine0lem  40568  ntrneineine1lem  40569  ntrneiel2  40571  clsneinex  40592  neicvgmex  40602
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