Theorem ntrclsneine0lem 38679

Description: If (pseudo-)interior and (pseudo-)closure functions are related by the duality operator then conditions equal to claiming that at least one (pseudo-)neighborbood of a particular point exists hold equally. (Contributed by RP, 21-May-2021.)

Hypotheses

Ref	Expression
ntrcls.o	⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
ntrcls.d	⊢ 𝐷 = (𝑂‘𝐵)
ntrcls.r	⊢ (𝜑 → 𝐼𝐷𝐾)
ntrclslem0.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)

Assertion

Ref	Expression
ntrclsneine0lem	⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾‘𝑠)))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑠   𝑗,𝐼,𝑘,𝑠   𝑋,𝑠   𝜑,𝑖,𝑗,𝑘,𝑠

Allowed substitution hints:   𝐷(𝑖,𝑗,𝑘,𝑠)   𝐼(𝑖)   𝐾(𝑖,𝑗,𝑘,𝑠)   𝑂(𝑖,𝑗,𝑘,𝑠)   𝑋(𝑖,𝑗,𝑘)

Proof of Theorem ntrclsneine0lem

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6229	. . . 4 ⊢ (𝑠 = 𝑡 → (𝐼‘𝑠) = (𝐼‘𝑡))
2	1	eleq2d 2716	. . 3 ⊢ (𝑠 = 𝑡 → (𝑋 ∈ (𝐼‘𝑠) ↔ 𝑋 ∈ (𝐼‘𝑡)))
3	2	cbvrexv 3202	. 2 ⊢ (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ ∃𝑡 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑡))
4		ntrcls.d	. . . . 5 ⊢ 𝐷 = (𝑂‘𝐵)
5		ntrcls.r	. . . . 5 ⊢ (𝜑 → 𝐼𝐷𝐾)
6	4, 5	ntrclsrcomplex 38650	. . . 4 ⊢ (𝜑 → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
7	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑠) ∈ 𝒫 𝐵)
8	4, 5	ntrclsrcomplex 38650	. . . . 5 ⊢ (𝜑 → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
9	8	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ∖ 𝑡) ∈ 𝒫 𝐵)
10		difeq2 3755	. . . . . 6 ⊢ (𝑠 = (𝐵 ∖ 𝑡) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑡)))
11	10	adantl 481	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑡)) → (𝐵 ∖ 𝑠) = (𝐵 ∖ (𝐵 ∖ 𝑡)))
12		elpwi 4201	. . . . . . 7 ⊢ (𝑡 ∈ 𝒫 𝐵 → 𝑡 ⊆ 𝐵)
13		dfss4 3891	. . . . . . 7 ⊢ (𝑡 ⊆ 𝐵 ↔ (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
14	12, 13	sylib 208	. . . . . 6 ⊢ (𝑡 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
15	14	ad2antlr 763	. . . . 5 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑡)) → (𝐵 ∖ (𝐵 ∖ 𝑡)) = 𝑡)
16	11, 15	eqtr2d 2686	. . . 4 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑠 = (𝐵 ∖ 𝑡)) → 𝑡 = (𝐵 ∖ 𝑠))
17	9, 16	rspcedeq2vd 3350	. . 3 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → ∃𝑠 ∈ 𝒫 𝐵𝑡 = (𝐵 ∖ 𝑠))
18		fveq2 6229	. . . . . 6 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → (𝐼‘𝑡) = (𝐼‘(𝐵 ∖ 𝑠)))
19	18	eleq2d 2716	. . . . 5 ⊢ (𝑡 = (𝐵 ∖ 𝑠) → (𝑋 ∈ (𝐼‘𝑡) ↔ 𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑠))))
20	19	3ad2ant3 1104	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝑋 ∈ (𝐼‘𝑡) ↔ 𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑠))))
21		ntrcls.o	. . . . . 6 ⊢ 𝑂 = (𝑖 ∈ V ↦ (𝑘 ∈ (𝒫 𝑖 ↑𝑚 𝒫 𝑖) ↦ (𝑗 ∈ 𝒫 𝑖 ↦ (𝑖 ∖ (𝑘‘(𝑖 ∖ 𝑗))))))
22	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝐼𝐷𝐾)
23		ntrclslem0.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
24	23	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑋 ∈ 𝐵)
25		simpr 476	. . . . . 6 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → 𝑠 ∈ 𝒫 𝐵)
26	21, 4, 22, 24, 25	ntrclselnel2 38673	. . . . 5 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑠)) ↔ ¬ 𝑋 ∈ (𝐾‘𝑠)))
27	26	3adant3 1101	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝑋 ∈ (𝐼‘(𝐵 ∖ 𝑠)) ↔ ¬ 𝑋 ∈ (𝐾‘𝑠)))
28	20, 27	bitrd 268	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵 ∧ 𝑡 = (𝐵 ∖ 𝑠)) → (𝑋 ∈ (𝐼‘𝑡) ↔ ¬ 𝑋 ∈ (𝐾‘𝑠)))
29	7, 17, 28	rexxfrd2 4915	. 2 ⊢ (𝜑 → (∃𝑡 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑡) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾‘𝑠)))
30	3, 29	syl5bb 272	1 ⊢ (𝜑 → (∃𝑠 ∈ 𝒫 𝐵𝑋 ∈ (𝐼‘𝑠) ↔ ∃𝑠 ∈ 𝒫 𝐵 ¬ 𝑋 ∈ (𝐾‘𝑠)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523   ∈ wcel 2030  ∃wrex 2942  Vcvv 3231   ∖ cdif 3604   ⊆ wss 3607  𝒫 cpw 4191   class class class wbr 4685   ↦ cmpt 4762  ‘cfv 5926  (class class class)co 6690   ↑_𝑚 cmap 7899

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-map 7901

This theorem is referenced by:  ntrclsneine0  38680