Theorem clsneiel1 44091

Description: If a (pseudo-)closure function and a (pseudo-)neighborhood function are related by the 𝐻 operator, then membership in the closure of a subset is equivalent to the complement of the subset not being a neighborhood of the point. (Contributed by RP, 7-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)
clsneiel.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
clsneiel.s	⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)

Assertion

Ref	Expression
clsneiel1	⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝑆) ↔ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋)))

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

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

Proof of Theorem clsneiel1

Step	Hyp	Ref	Expression
1		clsnei.d	. . . 4 ⊢ 𝐷 = (𝑃‘𝐵)
2		clsnei.h	. . . 4 ⊢ 𝐻 = (𝐹 ∘ 𝐷)
3		clsnei.r	. . . 4 ⊢ (𝜑 → 𝐾𝐻𝑁)
4	1, 2, 3	clsneibex 44085	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
5	4	ancli 548	. 2 ⊢ (𝜑 → (𝜑 ∧ 𝐵 ∈ V))
6		clsnei.o	. . . . 5 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
7		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
8	7	pwexd 5318	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
9		clsnei.f	. . . . 5 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
10	6, 8, 7, 9	fsovfd 43995	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)⟶(𝒫 𝒫 𝐵 ↑m 𝐵))
11	10	ffnd 6653	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐹 Fn (𝒫 𝐵 ↑m 𝒫 𝐵))
12		clsnei.p	. . . . 5 ⊢ 𝑃 = (𝑛 ∈ V ↦ (𝑝 ∈ (𝒫 𝑛 ↑m 𝒫 𝑛) ↦ (𝑜 ∈ 𝒫 𝑛 ↦ (𝑛 ∖ (𝑝‘(𝑛 ∖ 𝑜))))))
13	12, 1, 7	dssmapf1od 44004	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
14		f1of 6764	. . . 4 ⊢ (𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)⟶(𝒫 𝐵 ↑m 𝒫 𝐵))
15	13, 14	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)⟶(𝒫 𝐵 ↑m 𝒫 𝐵))
16	2	breqi 5098	. . . . 5 ⊢ (𝐾𝐻𝑁 ↔ 𝐾(𝐹 ∘ 𝐷)𝑁)
17	3, 16	sylib 218	. . . 4 ⊢ (𝜑 → 𝐾(𝐹 ∘ 𝐷)𝑁)
18	17	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐾(𝐹 ∘ 𝐷)𝑁)
19	11, 15, 18	brcoffn 44013	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁))
20		simprl 770	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → 𝐾𝐷(𝐷‘𝐾))
21		clsneiel.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
22	21	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → 𝑋 ∈ 𝐵)
23		clsneiel.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
24	23	ad2antrr 726	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → 𝑆 ∈ 𝒫 𝐵)
25	12, 1, 20, 22, 24	ntrclselnel1 44040	. . 3 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (𝑋 ∈ (𝐾‘𝑆) ↔ ¬ 𝑋 ∈ ((𝐷‘𝐾)‘(𝐵 ∖ 𝑆))))
26		simprr 772	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (𝐷‘𝐾)𝐹𝑁)
27		simplr 768	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → 𝐵 ∈ V)
28		difssd 4088	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (𝐵 ∖ 𝑆) ⊆ 𝐵)
29	27, 28	sselpwd 5267	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
30	6, 9, 26, 22, 29	ntrneiel 44064	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (𝑋 ∈ ((𝐷‘𝐾)‘(𝐵 ∖ 𝑆)) ↔ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋)))
31	30	notbid 318	. . 3 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (¬ 𝑋 ∈ ((𝐷‘𝐾)‘(𝐵 ∖ 𝑆)) ↔ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋)))
32	25, 31	bitrd 279	. 2 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (𝑋 ∈ (𝐾‘𝑆) ↔ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋)))
33	5, 19, 32	syl2anc2 585	1 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝑆) ↔ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3394  Vcvv 3436   ∖ cdif 3900  𝒫 cpw 4551   class class class wbr 5092   ↦ cmpt 5173   ∘ ccom 5623  ⟶wf 6478  –1-1-onto→wf1o 6481  ‘cfv 6482  (class class class)co 7349   ∈ cmpo 7351   ↑_m cmap 8753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5218  ax-sep 5235  ax-nul 5245  ax-pow 5304  ax-pr 5371  ax-un 7671

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3344  df-rab 3395  df-v 3438  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5174  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fun 6484  df-fn 6485  df-f 6486  df-f1 6487  df-fo 6488  df-f1o 6489  df-fv 6490  df-ov 7352  df-oprab 7353  df-mpo 7354  df-1st 7924  df-2nd 7925  df-map 8755

This theorem is referenced by:  clsneiel2  44092  clsneifv4  44094