Theorem ntrneik4 44754

Description: Idempotence of the interior function is equivalent to stating a set, 𝑠, is a neighborhood of a point, 𝑥 is equivalent to there existing a special neighborhood, 𝑢, of 𝑥 such that a point is an element of the special neighborhood if and only if 𝑠 is also a neighborhood of the point. (Contributed by RP, 11-Jul-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
ntrneik4	⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ ∃𝑢 ∈ (𝑁‘𝑥)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑠 ∈ (𝑁‘𝑦)))))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠,𝑥,𝑦   𝑘,𝐼,𝑙,𝑚,𝑥,𝑦   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠,𝑥   𝑢,𝐵,𝑠,𝑥,𝑦   𝑢,𝑁,𝑦   𝜑,𝑢,𝑦

Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑥,𝑦,𝑢,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝐼(𝑢,𝑖,𝑗,𝑠)   𝑁(𝑥,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)   𝑂(𝑥,𝑦,𝑢,𝑖,𝑗,𝑘,𝑚,𝑠,𝑙)

Proof of Theorem ntrneik4

Step	Hyp	Ref	Expression
1		ntrnei.o	. . 3 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
2		ntrnei.f	. . 3 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
3		ntrnei.r	. . 3 ⊢ (𝜑 → 𝐼𝐹𝑁)
4	1, 2, 3	ntrneik4w 44753	. 2 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥))))
5	3	ad2antrr 738	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐼𝐹𝑁)
6		simplr 780	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑥 ∈ 𝐵)
7	1, 2, 3	ntrneiiex 44729	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
8		elmapi 8846	. . . . . . . . . 10 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
9	7, 8	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
10	9	ffvelcdmda 7080	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ∈ 𝒫 𝐵)
11	10	adantlr 727	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ∈ 𝒫 𝐵)
12	1, 2, 5, 6, 11	ntrneiel 44734	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑥 ∈ (𝐼‘(𝐼‘𝑠)) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥)))
13		simpr 489	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
14	1, 2, 5, 6, 13	ntrneiel2 44739	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑥 ∈ (𝐼‘(𝐼‘𝑠)) ↔ ∃𝑢 ∈ (𝑁‘𝑥)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑠 ∈ (𝑁‘𝑦))))
15	12, 14	bitr3d 284	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑠 ∈ 𝒫 𝐵) → ((𝐼‘𝑠) ∈ (𝑁‘𝑥) ↔ ∃𝑢 ∈ (𝑁‘𝑥)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑠 ∈ (𝑁‘𝑦))))
16	15	bibi2d 345	. . . 4 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵) ∧ 𝑠 ∈ 𝒫 𝐵) → ((𝑠 ∈ (𝑁‘𝑥) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥)) ↔ (𝑠 ∈ (𝑁‘𝑥) ↔ ∃𝑢 ∈ (𝑁‘𝑥)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑠 ∈ (𝑁‘𝑦)))))
17	16	ralbidva 3192	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥)) ↔ ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ ∃𝑢 ∈ (𝑁‘𝑥)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑠 ∈ (𝑁‘𝑦)))))
18	17	ralbidva 3192	. 2 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ (𝐼‘𝑠) ∈ (𝑁‘𝑥)) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ ∃𝑢 ∈ (𝑁‘𝑥)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑠 ∈ (𝑁‘𝑦)))))
19	4, 18	bitrd 282	1 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵(𝐼‘(𝐼‘𝑠)) = (𝐼‘𝑠) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵(𝑠 ∈ (𝑁‘𝑥) ↔ ∃𝑢 ∈ (𝑁‘𝑥)∀𝑦 ∈ 𝐵 (𝑦 ∈ 𝑢 ↔ 𝑠 ∈ (𝑁‘𝑦)))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  {crab 3422  Vcvv 3461  𝒫 cpw 4565   class class class wbr 5111   ↦ cmpt 5194  ⟶wf 6533  ‘cfv 6537  (class class class)co 7411   ∈ cmpo 7413   ↑_m cmap 8824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-map 8826

This theorem is referenced by: (None)