Theorem clsneiel1 45215

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 45209	. . 3 ⊢ (𝜑 → 𝐵 ∈ V)
5	4	ancli 558	. 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 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐵 ∈ V)
8	7	pwexd 5340	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝒫 𝐵 ∈ V)
9		clsnei.f	. . . . 5 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
10	6, 8, 7, 9	fsovfd 45119	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐹:(𝒫 𝐵 ↑m 𝒫 𝐵)⟶(𝒫 𝒫 𝐵 ↑m 𝐵))
11	10	ffnd 6713	. . 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 45128	. . . 4 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝒫 𝐵))
14		f1of 6827	. . . 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 18	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐷:(𝒫 𝐵 ↑m 𝒫 𝐵)⟶(𝒫 𝐵 ↑m 𝒫 𝐵))
16	2	breqi 5108	. . . . 5 ⊢ (𝐾𝐻𝑁 ↔ 𝐾(𝐹 ∘ 𝐷)𝑁)
17	3, 16	sylib 221	. . . 4 ⊢ (𝜑 → 𝐾(𝐹 ∘ 𝐷)𝑁)
18	17	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → 𝐾(𝐹 ∘ 𝐷)𝑁)
19	11, 15, 18	brcoffn 45137	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ V) → (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁))
20		simprl 783	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → 𝐾𝐷(𝐷‘𝐾))
21		clsneiel.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
22	21	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → 𝑋 ∈ 𝐵)
23		clsneiel.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝒫 𝐵)
24	23	ad2antrr 739	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → 𝑆 ∈ 𝒫 𝐵)
25	12, 1, 20, 22, 24	ntrclselnel1 45164	. . 3 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (𝑋 ∈ (𝐾‘𝑆) ↔ ¬ 𝑋 ∈ ((𝐷‘𝐾)‘(𝐵 ∖ 𝑆))))
26		simprr 785	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (𝐷‘𝐾)𝐹𝑁)
27		simplr 781	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → 𝐵 ∈ V)
28		difssd 4072	. . . . . 6 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (𝐵 ∖ 𝑆) ⊆ 𝐵)
29	27, 28	sselpwd 5289	. . . . 5 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (𝐵 ∖ 𝑆) ∈ 𝒫 𝐵)
30	6, 9, 26, 22, 29	ntrneiel 45188	. . . 4 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (𝑋 ∈ ((𝐷‘𝐾)‘(𝐵 ∖ 𝑆)) ↔ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋)))
31	30	notbid 321	. . 3 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (¬ 𝑋 ∈ ((𝐷‘𝐾)‘(𝐵 ∖ 𝑆)) ↔ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋)))
32	25, 31	bitrd 282	. 2 ⊢ (((𝜑 ∧ 𝐵 ∈ V) ∧ (𝐾𝐷(𝐷‘𝐾) ∧ (𝐷‘𝐾)𝐹𝑁)) → (𝑋 ∈ (𝐾‘𝑆) ↔ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋)))
33	5, 19, 32	syl2anc2 597	1 ⊢ (𝜑 → (𝑋 ∈ (𝐾‘𝑆) ↔ ¬ (𝐵 ∖ 𝑆) ∈ (𝑁‘𝑋)))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {crab 3412  Vcvv 3450   ∖ cdif 4044  𝒫 cpw 4556   class class class wbr 5102   ↦ cmpt 5185   ∘ ccom 5654  ⟶wf 6537  –1-1-onto→wf1o 6540  ‘cfv 6541  (class class class)co 7422   ∈ cmpo 7424   ↑_m cmap 8852

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7753

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7425  df-oprab 7426  df-mpo 7427  df-1st 8003  df-2nd 8004  df-map 8854

This theorem is used by:  clsneiel2  45216  clsneifv4  45218