Theorem ntrneiiex 45268

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 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
ntrnei.f	⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
ntrnei.r	⊢ (𝜑 → 𝐼𝐹𝑁)

Assertion

Ref	Expression
ntrneiiex	⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 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

Step	Hyp	Ref	Expression
1		ntrnei.o	. . . . 5 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
2		ntrnei.f	. . . . 5 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
3		ntrnei.r	. . . . 5 ⊢ (𝜑 → 𝐼𝐹𝑁)
4	1, 2, 3	ntrneif1o 45267	. . . 4 ⊢ (𝜑 → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)⟶1-1onto(𝒫 𝒫 𝐵 ↑m 𝐵))
5		f1orel 6849	. . . 4 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)⟶1-1onto(𝒫 𝒫 𝐵 ↑m 𝐵) → Rel 𝐹)
6	4, 5	syl 18	. . 3 ⊢ (𝜑 → Rel 𝐹)
7		releldm 5933	. . 3 ⊢ ((Rel 𝐹 ∧ 𝐼𝐹𝑁) → 𝐼 ∈ dom 𝐹)
8	6, 3, 7	syl2anc 596	. 2 ⊢ (𝜑 → 𝐼 ∈ dom 𝐹)
9		f1odm 6850	. . 3 ⊢ (𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)⟶1-1onto(𝒫 𝒫 𝐵 ↑m 𝐵) → dom 𝐹 = (𝒫 𝐵 ↑m 𝒫 𝐵))
10	4, 9	syl 18	. 2 ⊢ (𝜑 → dom 𝐹 = (𝒫 𝐵 ↑m 𝒫 𝐵))
11	8, 10	eleqtrd 2866	1 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454  𝒫 cpw 4560   class class class wbr 5106   ↦ cmpt 5189  dom cdm 5655  Rel wrel 5660  ⟶1-1ontowf1o 6557  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445   ↑_m cmap 8875

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-map 8877

This theorem is used by:  ntrneifv1  45271  ntrneifv2  45272  ntrneiel  45273  ntrneifv4  45277  ntrneiel2  45278  ntrneicls00  45281  ntrneicls11  45282  ntrneiiso  45283  ntrneik2  45284  ntrneikb  45286  ntrneixb  45287  ntrneik3  45288  ntrneix3  45289  ntrneik13  45290  ntrneix13  45291  ntrneik4w  45292  ntrneik4  45293