Theorem ntrneixb 44841

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 3952	. . . . . . . 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 44822	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
7		elmapi 8842	. . . . . . . . . . . . 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 7079	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ∈ 𝒫 𝐵)
10	9	elpwid 4571	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ⊆ 𝐵)
11	10	adantr 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ⊆ 𝐵)
12	8	ffvelcdmda 7079	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑡) ∈ 𝒫 𝐵)
13	12	elpwid 4571	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑡) ⊆ 𝐵)
14	13	adantlr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑡) ⊆ 𝐵)
15	11, 14	unssd 4145	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ⊆ 𝐵)
16	15	biantrurd 541	. . . . . . 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 3926	. . . . . . . . 9 ⊢ (𝐵 ⊆ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)))
18		elun 4107	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ (𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡)))
19	18	ralbii 3111	. . . . . . . . 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 3190	. . . . . 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 742	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐼𝐹𝑁)
27		simpr 489	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
28		simpllr 787	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑠 ∈ 𝒫 𝐵)
29	3, 4, 26, 27, 28	ntrneiel 44827	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑠 ∈ (𝑁‘𝑥)))
30		simplr 780	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑡 ∈ 𝒫 𝐵)
31	3, 4, 26, 27, 30	ntrneiel 44827	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑡) ↔ 𝑡 ∈ (𝑁‘𝑥)))
32	29, 31	orbi12d 931	. . . . . . 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 3186	. . . . 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 3186	. . 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 3186	. 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 3296	. 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)))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143  ∀wral 3079  {crab 3416  Vcvv 3455   ∪ cun 3903   ⊆ wss 3905  𝒫 cpw 4562   class class class wbr 5109   ↦ cmpt 5192  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412   ↑_m cmap 8820

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-map 8822

This theorem is referenced by: (None)