Theorem ntrneixb 45094

Description: The interiors (closures) of sets that span the base set also span the base set if and only if the neighborhoods (convergents) of every point contain at least one of every pair of sets that span the base set. (Contributed by RP, 11-Jun-2021.)

Hypotheses

Ref	Expression
ntrnei.o	⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
ntrnei.f	⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
ntrnei.r	⊢ (𝜑 → 𝐼𝐹𝑁)

Assertion

Ref	Expression
ntrneixb	⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) = 𝐵 → ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) = 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) = 𝐵 → (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))

Distinct variable groups:   𝐵,𝑖,𝑗,𝑘,𝑙,𝑚,𝑠,𝑡,𝑥   𝑘,𝐼,𝑙,𝑚,𝑥   𝜑,𝑖,𝑗,𝑘,𝑙,𝑠,𝑡,𝑥

Allowed substitution hints:   𝜑(𝑚)   𝐹(𝑥, 𝑡, 𝑖, 𝑗, 𝑘, 𝑚, 𝑠, 𝑙)   𝐼(𝑡, 𝑖, 𝑗, 𝑠)   𝑁(𝑥, 𝑡, 𝑖, 𝑗, 𝑘, 𝑚, 𝑠, 𝑙)   𝑂(𝑥, 𝑡, 𝑖, 𝑗, 𝑘, 𝑚, 𝑠, 𝑙)

Proof of Theorem ntrneixb

Step	Hyp	Ref	Expression
1		eqss 3946	. . . . . . . 8 ⊢ (((𝐼‘𝑠) ∪ (𝐼‘𝑡)) = 𝐵 ↔ (((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ⊆ 𝐵 ∧ 𝐵 ⊆ ((𝐼‘𝑠) ∪ (𝐼‘𝑡))))
2	1	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝐼‘𝑠) ∪ (𝐼‘𝑡)) = 𝐵 ↔ (((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ⊆ 𝐵 ∧ 𝐵 ⊆ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)))))
3		ntrnei.o	. . . . . . . . . . . . . 14 ⊢ 𝑂 = (𝑖 ∈ V, 𝑗 ∈ V ↦ (𝑘 ∈ (𝒫 𝑗 ↑m 𝑖) ↦ (𝑙 ∈ 𝑗 ↦ {𝑚 ∈ 𝑖 ∣ 𝑙 ∈ (𝑘‘𝑚)})))
4		ntrnei.f	. . . . . . . . . . . . . 14 ⊢ 𝐹 = (𝒫 𝐵𝑂𝐵)
5		ntrnei.r	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼𝐹𝑁)
6	3, 4, 5	ntrneiiex 45075	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
7		elmapi 8869	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
8	6, 7	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
9	8	ffvelcdmda 7084	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ∈ 𝒫 𝐵)
10	9	elpwid 4566	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ⊆ 𝐵)
11	10	adantr 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ⊆ 𝐵)
12	8	ffvelcdmda 7084	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑡) ∈ 𝒫 𝐵)
13	12	elpwid 4566	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑡) ⊆ 𝐵)
14	13	adantlr 728	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑡) ⊆ 𝐵)
15	11, 14	unssd 4138	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ⊆ 𝐵)
16	15	biantrurd 542	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ⊆ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ (((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ⊆ 𝐵 ∧ 𝐵 ⊆ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)))))
17		dfss3 3920	. . . . . . . . 9 ⊢ (𝐵 ⊆ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)))
18		elun 4100	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ (𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡)))
19	18	ralbii 3109	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡)))
20	17, 19	bitri 278	. . . . . . . 8 ⊢ (𝐵 ⊆ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡)))
21	20	a1i 11	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐵 ⊆ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡))))
22	2, 16, 21	3bitr2d 310	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝐼‘𝑠) ∪ (𝐼‘𝑡)) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡))))
23	22	imbi2d 343	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝑠 ∪ 𝑡) = 𝐵 → ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) = 𝐵) ↔ ((𝑠 ∪ 𝑡) = 𝐵 → ∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡)))))
24		r19.21v 3188	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐵 ((𝑠 ∪ 𝑡) = 𝐵 → (𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡))) ↔ ((𝑠 ∪ 𝑡) = 𝐵 → ∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡))))
25	24	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥 ∈ 𝐵 ((𝑠 ∪ 𝑡) = 𝐵 → (𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡))) ↔ ((𝑠 ∪ 𝑡) = 𝐵 → ∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡)))))
26	5	ad3antrrr 743	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐼𝐹𝑁)
27		simpr 490	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
28		simpllr 788	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑠 ∈ 𝒫 𝐵)
29	3, 4, 26, 27, 28	ntrneiel 45080	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑠 ∈ (𝑁‘𝑥)))
30		simplr 781	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑡 ∈ 𝒫 𝐵)
31	3, 4, 26, 27, 30	ntrneiel 45080	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑡) ↔ 𝑡 ∈ (𝑁‘𝑥)))
32	29, 31	orbi12d 932	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡)) ↔ (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥))))
33	32	imbi2d 343	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (((𝑠 ∪ 𝑡) = 𝐵 → (𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡))) ↔ ((𝑠 ∪ 𝑡) = 𝐵 → (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))
34	33	ralbidva 3184	. . . . 5 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (∀𝑥 ∈ 𝐵 ((𝑠 ∪ 𝑡) = 𝐵 → (𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡))) ↔ ∀𝑥 ∈ 𝐵 ((𝑠 ∪ 𝑡) = 𝐵 → (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))
35	23, 25, 34	3bitr2d 310	. . . 4 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝑠 ∪ 𝑡) = 𝐵 → ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) = 𝐵) ↔ ∀𝑥 ∈ 𝐵 ((𝑠 ∪ 𝑡) = 𝐵 → (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))
36	35	ralbidva 3184	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) = 𝐵 → ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) = 𝐵) ↔ ∀𝑡 ∈ 𝒫 𝐵∀𝑥 ∈ 𝐵 ((𝑠 ∪ 𝑡) = 𝐵 → (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))
37	36	ralbidva 3184	. 2 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) = 𝐵 → ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) = 𝐵) ↔ ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵∀𝑥 ∈ 𝐵 ((𝑠 ∪ 𝑡) = 𝐵 → (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))
38		ralrot3 3294	. 2 ⊢ (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵∀𝑥 ∈ 𝐵 ((𝑠 ∪ 𝑡) = 𝐵 → (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) = 𝐵 → (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥))))
39	37, 38	bitrdi 290	1 ⊢ (𝜑 → (∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) = 𝐵 → ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) = 𝐵) ↔ ∀𝑥 ∈ 𝐵 ∀𝑠 ∈ 𝒫 𝐵∀𝑡 ∈ 𝒫 𝐵((𝑠 ∪ 𝑡) = 𝐵 → (𝑠 ∈ (𝑁‘𝑥) ∨ 𝑡 ∈ (𝑁‘𝑥)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3077  {crab 3413  Vcvv 3451   ∪ cun 3897   ⊆ wss 3899  𝒫 cpw 4557   class class class wbr 5103   ↦ cmpt 5186  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422   ↑_m cmap 8847

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-map 8849

This theorem is used by: (None)