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Theorem ntrneinex 38894
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 38892 . . . 4 (𝜑𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵))
5 f1orel 6281 . . . 4 (𝐹:(𝒫 𝐵𝑚 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵𝑚 𝐵) → Rel 𝐹)
64, 5syl 17 . . 3 (𝜑 → Rel 𝐹)
7 relelrn 5497 . . 3 ((Rel 𝐹𝐼𝐹𝑁) → 𝑁 ∈ ran 𝐹)
86, 3, 7syl2anc 565 . 2 (𝜑𝑁 ∈ ran 𝐹)
9 dff1o2 6283 . . . 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 208 . . 3 (𝜑 → (𝐹 Fn (𝒫 𝐵𝑚 𝒫 𝐵) ∧ Fun 𝐹 ∧ ran 𝐹 = (𝒫 𝒫 𝐵𝑚 𝐵)))
1110simp3d 1137 . 2 (𝜑 → ran 𝐹 = (𝒫 𝒫 𝐵𝑚 𝐵))
128, 11eleqtrd 2851 1 (𝜑𝑁 ∈ (𝒫 𝒫 𝐵𝑚 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1070   = wceq 1630  wcel 2144  {crab 3064  Vcvv 3349  𝒫 cpw 4295   class class class wbr 4784  cmpt 4861  ccnv 5248  ran crn 5250  Rel wrel 5254  Fun wfun 6025   Fn wfn 6026  1-1-ontowf1o 6030  cfv 6031  (class class class)co 6792  cmpt2 6794  𝑚 cmap 8008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-map 8010
This theorem is referenced by:  ntrneifv2  38897  ntrneifv3  38899  ntrneineine0lem  38900  ntrneineine1lem  38901  ntrneiel2  38903  clsneinex  38924  neicvgmex  38934
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