Theorem ntrneiel 44529

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 6835	. . . . 5 ⊢ (𝑚 = 𝑆 → (𝐼‘𝑚) = (𝐼‘𝑆))
3	2	eleq2d 2823	. . . 4 ⊢ (𝑚 = 𝑆 → (𝑋 ∈ (𝐼‘𝑚) ↔ 𝑋 ∈ (𝐼‘𝑆)))
4	3	elrab3 3636	. . 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 44521	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
10	9	pwexd 5317	. . . . 5 ⊢ (𝜑 → 𝒫 𝐵 ∈ V)
11	6, 7, 8	ntrneiiex 44524	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
12		eqid 2737	. . . . 5 ⊢ (𝐹‘𝐼) = (𝐹‘𝐼)
13		ntrnei.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
14	6, 10, 9, 7, 11, 12, 13	fsovfvfvd 44459	. . . 4 ⊢ (𝜑 → ((𝐹‘𝐼)‘𝑋) = {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)})
15	6, 7, 8	ntrneifv1 44527	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐼) = 𝑁)
16	15	fveq1d 6837	. . . 4 ⊢ (𝜑 → ((𝐹‘𝐼)‘𝑋) = (𝑁‘𝑋))
17	14, 16	eqtr3d 2774	. . 3 ⊢ (𝜑 → {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)} = (𝑁‘𝑋))
18	17	eleq2d 2823	. 2 ⊢ (𝜑 → (𝑆 ∈ {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)} ↔ 𝑆 ∈ (𝑁‘𝑋)))
19	5, 18	bitr3d 281	1 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ 𝑆 ∈ (𝑁‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ∈ wcel 2114  {crab 3390  Vcvv 3430  𝒫 cpw 4542   class class class wbr 5086   ↦ cmpt 5167  ‘cfv 6493  (class class class)co 7361   ∈ cmpo 7363   ↑_m cmap 8767

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5303  ax-pr 5371  ax-un 7683

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-1st 7936  df-2nd 7937  df-map 8769

This theorem is referenced by:  ntrneifv3  44530  ntrneineine0lem  44531  ntrneineine1lem  44532  ntrneifv4  44533  ntrneiel2  44534  ntrneicls00  44537  ntrneicls11  44538  ntrneiiso  44539  ntrneik2  44540  ntrneix2  44541  ntrneikb  44542  ntrneixb  44543  ntrneik3  44544  ntrneix3  44545  ntrneik13  44546  ntrneix13  44547  ntrneik4w  44548  ntrneik4  44549  clsneiel1  44556  neicvgel1  44567