Theorem ntrneiel 39150

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 6410	. . . . 5 ⊢ (𝑚 = 𝑆 → (𝐼‘𝑚) = (𝐼‘𝑆))
3	2	eleq2d 2863	. . . 4 ⊢ (𝑚 = 𝑆 → (𝑋 ∈ (𝐼‘𝑚) ↔ 𝑋 ∈ (𝐼‘𝑆)))
4	3	elrab3 3557	. . 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 39142	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ V)
10	9	pwexd 5048	. . . . 5 ⊢ (𝜑 → 𝒫 𝐵 ∈ V)
11	6, 7, 8	ntrneiiex 39145	. . . . 5 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑𝑚 𝒫 𝐵))
12		eqid 2798	. . . . 5 ⊢ (𝐹‘𝐼) = (𝐹‘𝐼)
13		ntrnei.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
14	6, 10, 9, 7, 11, 12, 13	fsovfvfvd 39076	. . . 4 ⊢ (𝜑 → ((𝐹‘𝐼)‘𝑋) = {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)})
15	6, 7, 8	ntrneifv1 39148	. . . . 5 ⊢ (𝜑 → (𝐹‘𝐼) = 𝑁)
16	15	fveq1d 6412	. . . 4 ⊢ (𝜑 → ((𝐹‘𝐼)‘𝑋) = (𝑁‘𝑋))
17	14, 16	eqtr3d 2834	. . 3 ⊢ (𝜑 → {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)} = (𝑁‘𝑋))
18	17	eleq2d 2863	. 2 ⊢ (𝜑 → (𝑆 ∈ {𝑚 ∈ 𝒫 𝐵 ∣ 𝑋 ∈ (𝐼‘𝑚)} ↔ 𝑆 ∈ (𝑁‘𝑋)))
19	5, 18	bitr3d 273	1 ⊢ (𝜑 → (𝑋 ∈ (𝐼‘𝑆) ↔ 𝑆 ∈ (𝑁‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1653   ∈ wcel 2157  {crab 3092  Vcvv 3384  𝒫 cpw 4348   class class class wbr 4842   ↦ cmpt 4921  ‘cfv 6100  (class class class)co 6877   ↦ cmpt2 6879   ↑_𝑚 cmap 8094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-rep 4963  ax-sep 4974  ax-nul 4982  ax-pow 5034  ax-pr 5096  ax-un 7182

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-ral 3093  df-rex 3094  df-reu 3095  df-rab 3097  df-v 3386  df-sbc 3633  df-csb 3728  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4628  df-iun 4711  df-br 4843  df-opab 4905  df-mpt 4922  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6063  df-fun 6102  df-fn 6103  df-f 6104  df-f1 6105  df-fo 6106  df-f1o 6107  df-fv 6108  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-1st 7400  df-2nd 7401  df-map 8096

This theorem is referenced by:  ntrneifv3  39151  ntrneineine0lem  39152  ntrneineine1lem  39153  ntrneifv4  39154  ntrneiel2  39155  ntrneicls00  39158  ntrneicls11  39159  ntrneiiso  39160  ntrneik2  39161  ntrneix2  39162  ntrneikb  39163  ntrneixb  39164  ntrneik3  39165  ntrneix3  39166  ntrneik13  39167  ntrneix13  39168  ntrneik4w  39169  ntrneik4  39170  clsneiel1  39177  neicvgel1  39188