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Theorem ntrneiiex 41724
Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then the interior 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
ntrneiiex (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 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 ntrneiiex
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 41723 . . . 4 (𝜑𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵))
5 f1orel 6749 . . . 4 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) → Rel 𝐹)
64, 5syl 17 . . 3 (𝜑 → Rel 𝐹)
7 releldm 5865 . . 3 ((Rel 𝐹𝐼𝐹𝑁) → 𝐼 ∈ dom 𝐹)
86, 3, 7syl2anc 585 . 2 (𝜑𝐼 ∈ dom 𝐹)
9 f1odm 6750 . . 3 (𝐹:(𝒫 𝐵m 𝒫 𝐵)–1-1-onto→(𝒫 𝒫 𝐵m 𝐵) → dom 𝐹 = (𝒫 𝐵m 𝒫 𝐵))
104, 9syl 17 . 2 (𝜑 → dom 𝐹 = (𝒫 𝐵m 𝒫 𝐵))
118, 10eleqtrd 2839 1 (𝜑𝐼 ∈ (𝒫 𝐵m 𝒫 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1539  wcel 2104  {crab 3284  Vcvv 3437  𝒫 cpw 4539   class class class wbr 5081  cmpt 5164  dom cdm 5600  Rel wrel 5605  1-1-ontowf1o 6457  cfv 6458  (class class class)co 7307  cmpo 7309  m cmap 8646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-rep 5218  ax-sep 5232  ax-nul 5239  ax-pow 5297  ax-pr 5361  ax-un 7620
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3286  df-rab 3287  df-v 3439  df-sbc 3722  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-iun 4933  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-iota 6410  df-fun 6460  df-fn 6461  df-f 6462  df-f1 6463  df-fo 6464  df-f1o 6465  df-fv 6466  df-ov 7310  df-oprab 7311  df-mpo 7312  df-1st 7863  df-2nd 7864  df-map 8648
This theorem is referenced by:  ntrneifv1  41727  ntrneifv2  41728  ntrneiel  41729  ntrneifv4  41733  ntrneiel2  41734  ntrneicls00  41737  ntrneicls11  41738  ntrneiiso  41739  ntrneik2  41740  ntrneikb  41742  ntrneixb  41743  ntrneik3  41744  ntrneix3  41745  ntrneik13  41746  ntrneix13  41747  ntrneik4w  41748  ntrneik4  41749
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