Theorem clsneiel1 42844

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 42838	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
5	4	ancli 549	. 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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
8	7	pwexd 5376	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
9		clsnei.f	. . . . 5 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
10	6, 8, 7, 9	fsovfd 42748	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)⟶(𝒫 𝒫 𝐵 ↑m 𝐵))
11	10	ffnd 6715	. . 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 42757	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
14		f1of 6830	. . . 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 5153	. . . . 5 ⊢ (𝐾𝐻𝑁 ↔ 𝐾(𝐹 ∘ 𝐷)𝑁)
17	3, 16	sylib 217	. . . 4 ⊢ (𝜑 → 𝐾(𝐹 ∘ 𝐷)𝑁)
18	17	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐾(𝐹 ∘ 𝐷)𝑁)
19	11, 15, 18	brcoffn 42766	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁))
20		simprl 769	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → 𝐾𝐷(𝐷‘𝐾))
21		clsneiel.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
22	21	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → 𝑋 ∈ 𝐵)
23		clsneiel.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
24	23	ad2antrr 724	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → 𝑆 ∈ 𝒫 𝐵)
25	12, 1, 20, 22, 24	ntrclselnel1 42793	. . 3 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (𝑋 ∈ (𝐾‘𝑆) ↔ ¬ 𝑋 ∈ ((𝐷‘𝐾)‘(𝐵 ∖ 𝑆))))
26		simprr 771	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (𝐷‘𝐾)𝐹𝑁)
27		simplr 767	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → 𝐵 ∈ V)
28		difssd 4131	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (𝐵 ∖ 𝑆) ⊆ 𝐵)
29	27, 28	sselpwd 5325	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
30	6, 9, 26, 22, 29	ntrneiel 42817	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (𝑋 ∈ ((𝐷‘𝐾)‘(𝐵 ∖ 𝑆)) ↔ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋)))
31	30	notbid 317	. . 3 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (¬ 𝑋 ∈ ((𝐷‘𝐾)‘(𝐵 ∖ 𝑆)) ↔ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋)))
32	25, 31	bitrd 278	. 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 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  {crab 3432  Vcvv 3474   ∖ cdif 3944  𝒫 cpw 4601   class class class wbr 5147   ↦ cmpt 5230   ∘ ccom 5679  ⟶wf 6536  –1-1-onto→wf1o 6539  ‘cfv 6540  (class class class)co 7405   ∈ cmpo 7407   ↑_m cmap 8816

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-map 8818

This theorem is referenced by:  clsneiel2  42845  clsneifv4  42847