Theorem clsneifv4 45372

Description: Value of the closure (interior) function in terms of the neighborhoods (convergents) function. (Contributed by RP, 27-Jun-2021.)

Hypotheses

Ref	Expression
clsnei.o	⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
clsnei.p	⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))
clsnei.d	⊢ 𝐷 = (𝑃‘𝐵)
clsnei.f	⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
clsnei.h	⊢ 𝐻 = (𝐹 ∘ 𝐷)
clsnei.r	⊢ (𝜑 → 𝐾𝐻𝑁)
clsneifv.s	⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)

Assertion

Ref	Expression
clsneifv4	⊢ (𝜑 → (𝐾‘𝑆) = {𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)})

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑥   𝐵,𝑛,𝑜,𝑝,𝑥   𝐷,𝑖,𝑗,𝑘,𝑙,𝑚   𝐷,𝑛,𝑜,𝑝   𝑖,𝐹,𝑗,𝑘,𝑙   𝑛,𝐹,𝑜,𝑝   𝑖,𝐾,𝑗,𝑘,𝑙,𝑚,𝑥   𝑛,𝐾,𝑜,𝑝   𝑖,𝑁,𝑗,𝑘,𝑙   𝑛,𝑁,𝑜,𝑝   𝑆,𝑚,𝑥   𝑆,𝑜   𝜑,𝑖,𝑗,𝑘,𝑙,𝑥   𝜑,𝑛,𝑜,𝑝

Allowed substitution hints:   𝜑(𝑚)   𝐷(𝑥)   𝑃(𝑥, 𝑖, 𝑗, 𝑘, 𝑚, 𝑛, 𝑜, 𝑝, 𝑙)   𝑆(𝑖, 𝑗, 𝑘, 𝑛, 𝑝, 𝑙)   𝐹(𝑥, 𝑚)   𝐻(𝑥, 𝑖, 𝑗, 𝑘, 𝑚, 𝑛, 𝑜, 𝑝, 𝑙)   𝑁(𝑥, 𝑚)   𝑂(𝑥, 𝑖, 𝑗, 𝑘, 𝑚, 𝑛, 𝑜, 𝑝, 𝑙)

Proof of Theorem clsneifv4

Step	Hyp	Ref	Expression
1		dfin5 4095	. 2 ⊢ (𝐵 ∩ (𝐾‘𝑆)) = {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ (𝐾‘𝑆)}
2		clsnei.o	. . . . . . 7 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
3		clsnei.p	. . . . . . 7 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))
4		clsnei.d	. . . . . . 7 ⊢ 𝐷 = (𝑃‘𝐵)
5		clsnei.f	. . . . . . 7 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
6		clsnei.h	. . . . . . 7 ⊢ 𝐻 = (𝐹 ∘ 𝐷)
7		clsnei.r	. . . . . . 7 ⊢ (𝜑 → 𝐾𝐻𝑁)
8	2, 3, 4, 5, 6, 7	clsneikex 45367	. . . . . 6 ⊢ (𝜑 → 𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
9		elmapi 8907	. . . . . 6 ⊢ (𝐾 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
10	8, 9	syl 18	. . . . 5 ⊢ (𝜑 → 𝐾:𝒫 𝐵⟶𝒫 𝐵)
11		clsneifv.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
12	10, 11	ffvelcdmd 7117	. . . 4 ⊢ (𝜑 → (𝐾‘𝑆) ∈ 𝒫 𝐵)
13	12	elpwid 4569	. . 3 ⊢ (𝜑 → (𝐾‘𝑆) ⊆ 𝐵)
14		sseqin2 4123	. . 3 ⊢ ((𝐾‘𝑆) ⊆ 𝐵 ↔ (𝐵 ∩ (𝐾‘𝑆)) = (𝐾‘𝑆))
15	13, 14	sylib 221	. 2 ⊢ (𝜑 → (𝐵 ∩ (𝐾‘𝑆)) = (𝐾‘𝑆))
16	7	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝐾𝐻𝑁)
17		simpr 490	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
18	11	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → 𝑆 ∈ 𝒫 𝐵)
19	2, 3, 4, 5, 6, 16, 17, 18	clsneiel1 45369	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐾‘𝑆) ↔ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)))
20	19	rabbidva 3422	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ 𝑥 ∈ (𝐾‘𝑆)} = {𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)})
21	1, 15, 20	3eqtr3a 2822	1 ⊢ (𝜑 → (𝐾‘𝑆) = {𝑥 ∈ 𝐵 ∣ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑥)})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092  𝒫 cpw 4560   class class class wbr 5108   ↦ cmpt 5192   ∘ ccom 5662  ⟶wf 6562  ‘cfv 6566  (class class class)co 7451   ∈ cmpo 7453   ↑_m cmap 8885

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 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-1st 8035  df-2nd 8036  df-map 8887

This theorem is used by: (None)