Theorem ntrneiel 41580

Description: If (pseudo-)interior and (pseudo-)neighborhood functions are related by the operator, 𝐹, then there is an equivalence between membership in the interior of a set and non-membership in the closure of the complement of the set. (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)
ntrnei.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ntrnei.s	⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)

Assertion

Ref	Expression
ntrneiel	⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ 𝑆 ∈ (𝑁‘𝑋)))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚   𝑘,𝐼,𝑙,𝑚   𝑆,𝑚   𝑋,𝑙,𝑚   𝜑,𝑖,𝑗,𝑘,𝑙

Allowed substitution hints:   𝜑(𝑚)   𝑆(𝑖,𝑗,𝑘,𝑙)   𝐹(𝑖,𝑗,𝑘,𝑚,𝑙)   𝐼(𝑖,𝑗)   𝑁(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑂(𝑖,𝑗,𝑘,𝑚,𝑙)   𝑋(𝑖,𝑗,𝑘)

Proof of Theorem ntrneiel

Step	Hyp	Ref	Expression
1		ntrnei.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
2		fveq2 6756	. . . . 5 ⊢ (𝑚 = 𝑆 → (𝐼‘𝑚) = (𝐼‘𝑆))
3	2	eleq2d 2824	. . . 4 ⊢ (𝑚 = 𝑆 → (𝑋 ∈ (𝐼‘𝑚) ↔ 𝑋 ∈ (𝐼‘𝑆)))
4	3	elrab3 3618	. . 3 ⊢ (𝑆 ∈ 𝒫 𝐵 → (𝑆 ∈ {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)} ↔ 𝑋 ∈ (𝐼‘𝑆)))
5	1, 4	syl 17	. 2 ⊢ (𝜑 → (𝑆 ∈ {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)} ↔ 𝑋 ∈ (𝐼‘𝑆)))
6		ntrnei.o	. . . . 5 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
7		ntrnei.f	. . . . . . 7 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
8		ntrnei.r	. . . . . . 7 ⊢ (𝜑 → 𝐼𝐹𝑁)
9	6, 7, 8	ntrneibex 41572	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
10	9	pwexd 5297	. . . . 5 ⊢ (𝜑 → 𝒫 𝐵 ∈ V)
11	6, 7, 8	ntrneiiex 41575	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
12		eqid 2738	. . . . 5 ⊢ (𝐹‘𝐼) = (𝐹‘𝐼)
13		ntrnei.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
14	6, 10, 9, 7, 11, 12, 13	fsovfvfvd 41508	. . . 4 ⊢ (𝜑 → ((𝐹‘𝐼)‘𝑋) = {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)})
15	6, 7, 8	ntrneifv1 41578	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐼) = 𝑁)
16	15	fveq1d 6758	. . . 4 ⊢ (𝜑 → ((𝐹‘𝐼)‘𝑋) = (𝑁‘𝑋))
17	14, 16	eqtr3d 2780	. . 3 ⊢ (𝜑 → {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)} = (𝑁‘𝑋))
18	17	eleq2d 2824	. 2 ⊢ (𝜑 → (𝑆 ∈ {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)} ↔ 𝑆 ∈ (𝑁‘𝑋)))
19	5, 18	bitr3d 280	1 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ 𝑆 ∈ (𝑁‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108  {crab 3067  Vcvv 3422  𝒫 cpw 4530   class class class wbr 5070   ↦ cmpt 5153  ‘cfv 6418  (class class class)co 7255   ∈ cmpo 7257   ↑_m cmap 8573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-map 8575

This theorem is referenced by:  ntrneifv3  41581  ntrneineine0lem  41582  ntrneineine1lem  41583  ntrneifv4  41584  ntrneiel2  41585  ntrneicls00  41588  ntrneicls11  41589  ntrneiiso  41590  ntrneik2  41591  ntrneix2  41592  ntrneikb  41593  ntrneixb  41594  ntrneik3  41595  ntrneix3  41596  ntrneik13  41597  ntrneix13  41598  ntrneik4w  41599  ntrneik4  41600  clsneiel1  41607  neicvgel1  41618