Theorem ntrneixb 44091

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 3965	. . . . . . . 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 44072	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵))
7		elmapi 8825	. . . . . . . . . . . . 13 ⊢ (𝐼 ∈ (𝒫 𝐵 ↑m 𝒫 𝐵) → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
8	6, 7	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐼:𝒫 𝐵⟶𝒫 𝐵)
9	8	ffvelcdmda 7059	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ∈ 𝒫 𝐵)
10	9	elpwid 4575	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ⊆ 𝐵)
11	10	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑠) ⊆ 𝐵)
12	8	ffvelcdmda 7059	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑡) ∈ 𝒫 𝐵)
13	12	elpwid 4575	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑡) ⊆ 𝐵)
14	13	adantlr 715	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (𝐼‘𝑡) ⊆ 𝐵)
15	11, 14	unssd 4158	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ⊆ 𝐵)
16	15	biantrurd 532	. . . . . . 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 3938	. . . . . . . . 9 ⊢ (𝐵 ⊆ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)))
18		elun 4119	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ (𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡)))
19	18	ralbii 3076	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐵 𝑥 ∈ ((𝐼‘𝑠) ∪ (𝐼‘𝑡)) ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡)))
20	17, 19	bitri 275	. . . . . . . 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 307	. . . . . 6 ⊢ (((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) → (((𝐼‘𝑠) ∪ (𝐼‘𝑡)) = 𝐵 ↔ ∀𝑥 ∈ 𝐵 (𝑥 ∈ (𝐼‘𝑠) ∨ 𝑥 ∈ (𝐼‘𝑡))))
23	22	imbi2d 340	. . . . 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 3159	. . . . . 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 730	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝐼𝐹𝑁)
27		simpr 484	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑥 ∈ 𝐵)
28		simpllr 775	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑠 ∈ 𝒫 𝐵)
29	3, 4, 26, 27, 28	ntrneiel 44077	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑠) ↔ 𝑠 ∈ (𝑁‘𝑥)))
30		simplr 768	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → 𝑡 ∈ 𝒫 𝐵)
31	3, 4, 26, 27, 30	ntrneiel 44077	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑠 ∈ 𝒫 𝐵) ∧ 𝑡 ∈ 𝒫 𝐵) ∧ 𝑥 ∈ 𝐵) → (𝑥 ∈ (𝐼‘𝑡) ↔ 𝑡 ∈ (𝑁‘𝑥)))
32	29, 31	orbi12d 918	. . . . . . 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 340	. . . . . 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 3155	. . . . 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 307	. . . 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 3155	. . 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 3155	. 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 3270	. 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 287	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 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  ∀wral 3045  {crab 3408  Vcvv 3450   ∪ cun 3915   ⊆ wss 3917  𝒫 cpw 4566   class class class wbr 5110   ↦ cmpt 5191  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392   ↑_m cmap 8802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-map 8804

This theorem is referenced by: (None)