Theorem prdsxmslem2 22334

Description: Lemma for prdsxms 22335. The topology generated by the supremum metric is the same as the product topology, when the index set is finite. (Contributed by Mario Carneiro, 28-Aug-2015.)

Hypotheses

Ref	Expression
prdsxms.y	⊢ 𝑌 = (𝑆Xs𝑅)
prdsxms.s	⊢ (𝜑 → 𝑆 ∈ 𝑊)
prdsxms.i	⊢ (𝜑 → 𝐼 ∈ Fin)
prdsxms.d	⊢ 𝐷 = (dist‘𝑌)
prdsxms.b	⊢ 𝐵 = (Base‘𝑌)
prdsxms.r	⊢ (𝜑 → 𝑅:𝐼⟶∞MetSp)
prdsxms.j	⊢ 𝐽 = (TopOpen‘𝑌)
prdsxms.v	⊢ 𝑉 = (Base‘(𝑅‘𝑘))
prdsxms.e	⊢ 𝐸 = ((dist‘(𝑅‘𝑘)) ↾ (𝑉 × 𝑉))
prdsxms.k	⊢ 𝐾 = (TopOpen‘(𝑅‘𝑘))
prdsxms.c	⊢ 𝐶 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))}

Assertion

Ref	Expression
prdsxmslem2	⊢ (𝜑 → 𝐽 = (MetOpen‘𝐷))

Distinct variable groups:   𝑔,𝑘,𝐵   𝑥,𝑔,𝐷,𝑘   𝑧,𝑔,𝐼,𝑘,𝑥   𝑔,𝐸   𝑆,𝑔,𝑘,𝑥   𝑔,𝑊,𝑘,𝑥   𝑔,𝑌,𝑘,𝑥   𝜑,𝑔,𝑘,𝑥   𝑅,𝑔,𝑘,𝑥,𝑧

Allowed substitution hints:   𝜑(𝑧)   𝐵(𝑥,𝑧)   𝐶(𝑥,𝑧,𝑔,𝑘)   𝐷(𝑧)   𝑆(𝑧)   𝐸(𝑥,𝑧,𝑘)   𝐽(𝑥,𝑧,𝑔,𝑘)   𝐾(𝑥,𝑧,𝑔,𝑘)   𝑉(𝑥,𝑧,𝑔,𝑘)   𝑊(𝑧)   𝑌(𝑧)

Proof of Theorem prdsxmslem2

Dummy variables 𝑝 𝑟 𝑤 𝑦 𝑚 𝑢 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		prdsxms.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ Fin)
2		topnfn 16086	. . . . 5 ⊢ TopOpen Fn V
3		prdsxms.r	. . . . . . 7 ⊢ (𝜑 → 𝑅:𝐼⟶∞MetSp)
4		ffn 6045	. . . . . . 7 ⊢ (𝑅:𝐼⟶∞MetSp → 𝑅 Fn 𝐼)
5	3, 4	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 Fn 𝐼)
6		dffn2 6047	. . . . . 6 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
7	5, 6	sylib 208	. . . . 5 ⊢ (𝜑 → 𝑅:𝐼⟶V)
8		fnfco 6069	. . . . 5 ⊢ ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
9	2, 7, 8	sylancr 695	. . . 4 ⊢ (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
10		prdsxms.c	. . . . 5 ⊢ 𝐶 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))}
11	10	ptval 21373	. . . 4 ⊢ ((𝐼 ∈ Fin ∧ (TopOpen ∘ 𝑅) Fn 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘𝐶))
12	1, 9, 11	syl2anc 693	. . 3 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘𝐶))
13		eldifsn 4317	. . . . . . . 8 ⊢ (𝑥 ∈ (ran (ball‘𝐷) ∖ {∅}) ↔ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑥 ≠ ∅))
14		prdsxms.y	. . . . . . . . . . . 12 ⊢ 𝑌 = (𝑆Xs𝑅)
15		prdsxms.s	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
16		prdsxms.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (dist‘𝑌)
17		prdsxms.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑌)
18	14, 15, 1, 16, 17, 3	prdsxmslem1 22333	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))
19		blrn 22214	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (∞Met‘𝐵) → (𝑥 ∈ ran (ball‘𝐷) ↔ ∃𝑝 ∈ 𝐵 ∃𝑟 ∈ ℝ* 𝑥 = (𝑝(ball‘𝐷)𝑟)))
20	18, 19	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ran (ball‘𝐷) ↔ ∃𝑝 ∈ 𝐵 ∃𝑟 ∈ ℝ* 𝑥 = (𝑝(ball‘𝐷)𝑟)))
21	18	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → 𝐷 ∈ (∞Met‘𝐵))
22		simprl 794	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → 𝑝 ∈ 𝐵)
23		simprr 796	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → 𝑟 ∈ ℝ*)
24		xbln0 22219	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
25	21, 22, 23, 24	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
26	1	3ad2ant1 1082	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐼 ∈ Fin)
27		mptexg 6484	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐼 ∈ Fin → (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) ∈ V)
28	26, 27	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) ∈ V)
29		ovex 6678	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟) ∈ V
30	29	rgenw 2924	. . . . . . . . . . . . . . . . . 18 ⊢ ∀𝑛 ∈ 𝐼 ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟) ∈ V
31		eqid 2622	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))
32	31	fnmpt 6020	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑛 ∈ 𝐼 ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟) ∈ V → (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) Fn 𝐼)
33	30, 32	mp1i 13	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) Fn 𝐼)
34	3	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅:𝐼⟶∞MetSp)
35	34	ffvelrnda 6359	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ ∞MetSp)
36		prdsxms.v	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑉 = (Base‘(𝑅‘𝑘))
37		prdsxms.e	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐸 = ((dist‘(𝑅‘𝑘)) ↾ (𝑉 × 𝑉))
38	36, 37	xmsxmet 22261	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅‘𝑘) ∈ ∞MetSp → 𝐸 ∈ (∞Met‘𝑉))
39	35, 38	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
40		eqid 2622	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))) = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))
41		eqid 2622	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))
42	15	3ad2ant1 1082	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑆 ∈ 𝑊)
43	35	ralrimiva 2966	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∀𝑘 ∈ 𝐼 (𝑅‘𝑘) ∈ ∞MetSp)
44		simp2l 1087	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝 ∈ 𝐵)
45	34	feqmptd 6249	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅 = (𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))
46	45	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑆Xs𝑅) = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))
47	14, 46	syl5eq 2668	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑌 = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))
48	47	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (Base‘𝑌) = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))))
49	17, 48	syl5eq 2668	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐵 = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))))
50	44, 49	eleqtrd 2703	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝 ∈ (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))))
51	40, 41, 42, 26, 43, 36, 50	prdsbascl 16143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∀𝑘 ∈ 𝐼 (𝑝‘𝑘) ∈ 𝑉)
52	51	r19.21bi 2932	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → (𝑝‘𝑘) ∈ 𝑉)
53		simp2r 1088	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑟 ∈ ℝ*)
54	53	adantr 481	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → 𝑟 ∈ ℝ*)
55		eqid 2622	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (MetOpen‘𝐸) = (MetOpen‘𝐸)
56	55	blopn 22305	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑝‘𝑘) ∈ 𝑉 ∧ 𝑟 ∈ ℝ*) → ((𝑝‘𝑘)(ball‘𝐸)𝑟) ∈ (MetOpen‘𝐸))
57	39, 52, 54, 56	syl3anc 1326	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → ((𝑝‘𝑘)(ball‘𝐸)𝑟) ∈ (MetOpen‘𝐸))
58		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑘 → (𝑅‘𝑛) = (𝑅‘𝑘))
59	58	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑘 → (dist‘(𝑅‘𝑛)) = (dist‘(𝑅‘𝑘)))
60	58	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = 𝑘 → (Base‘(𝑅‘𝑛)) = (Base‘(𝑅‘𝑘)))
61	60, 36	syl6eqr 2674	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑘 → (Base‘(𝑅‘𝑛)) = 𝑉)
62	61	sqxpeqd 5141	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑘 → ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛))) = (𝑉 × 𝑉))
63	59, 62	reseq12d 5397	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑘 → ((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))) = ((dist‘(𝑅‘𝑘)) ↾ (𝑉 × 𝑉)))
64	63, 37	syl6eqr 2674	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑘 → ((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))) = 𝐸)
65	64	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑘 → (ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛))))) = (ball‘𝐸))
66		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑘 → (𝑝‘𝑛) = (𝑝‘𝑘))
67		eqidd 2623	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑘 → 𝑟 = 𝑟)
68	65, 66, 67	oveq123d 6671	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑘 → ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟) = ((𝑝‘𝑘)(ball‘𝐸)𝑟))
69		ovex 6678	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ∈ V
70	68, 31, 69	fvmpt 6282	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝐼 → ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) = ((𝑝‘𝑘)(ball‘𝐸)𝑟))
71	70	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) = ((𝑝‘𝑘)(ball‘𝐸)𝑟))
72		fvco3 6275	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅:𝐼⟶∞MetSp ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅‘𝑘)))
73		prdsxms.k	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐾 = (TopOpen‘(𝑅‘𝑘))
74	72, 73	syl6eqr 2674	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅:𝐼⟶∞MetSp ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
75	34, 74	sylan 488	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
76	73, 36, 37	xmstopn 22256	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅‘𝑘) ∈ ∞MetSp → 𝐾 = (MetOpen‘𝐸))
77	35, 76	syl 17	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → 𝐾 = (MetOpen‘𝐸))
78	75, 77	eqtrd 2656	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (MetOpen‘𝐸))
79	57, 71, 78	3eltr4d 2716	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘))
80	79	ralrimiva 2966	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∀𝑘 ∈ 𝐼 ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘))
81	34	feqmptd 6249	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅 = (𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))
82	81	oveq2d 6666	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑆Xs𝑅) = (𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))
83	14, 82	syl5eq 2668	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑌 = (𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))
84	83	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (dist‘𝑌) = (dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))
85	16, 84	syl5eq 2668	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐷 = (dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))
86	85	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (ball‘𝐷) = (ball‘(dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))))
87	86	oveqd 6667	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘𝐷)𝑟) = (𝑝(ball‘(dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))𝑟))
88	58	cbvmptv 4750	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)) = (𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))
89	88	oveq2i 6661	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))) = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))
90		eqid 2622	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))) = (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))
91		eqid 2622	. . . . . . . . . . . . . . . . . . 19 ⊢ (dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))) = (dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))
92	83	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (Base‘𝑌) = (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))
93	17, 92	syl5eq 2668	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐵 = (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))
94	44, 93	eleqtrd 2703	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝 ∈ (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))
95		simp3 1063	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 0 < 𝑟)
96	89, 90, 36, 37, 91, 42, 26, 35, 39, 94, 53, 95	prdsbl 22296	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘(dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟))
97	87, 96	eqtrd 2656	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟))
98		fneq1 5979	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) → (𝑔 Fn 𝐼 ↔ (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) Fn 𝐼))
99		fveq1 6190	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) → (𝑔‘𝑘) = ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘))
100	99	eleq1d 2686	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) → ((𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ↔ ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)))
101	100	ralbidv 2986	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) → (∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ↔ ∀𝑘 ∈ 𝐼 ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)))
102	98, 101	anbi12d 747	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) → ((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ↔ ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘))))
103	99, 70	sylan9eq 2676	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) ∧ 𝑘 ∈ 𝐼) → (𝑔‘𝑘) = ((𝑝‘𝑘)(ball‘𝐸)𝑟))
104	103	ixpeq2dva 7923	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) → X𝑘 ∈ 𝐼 (𝑔‘𝑘) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟))
105	104	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) → ((𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘) ↔ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟)))
106	102, 105	anbi12d 747	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑔 = (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) → (((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘)) ↔ (((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟))))
107	106	spcegv 3294	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) ∈ V → ((((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟)) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘))))
108	107	3impib 1262	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) ∈ V ∧ ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟)) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))
109	28, 33, 80, 97, 108	syl121anc 1331	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))
110	109	3expia 1267	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → (0 < 𝑟 → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘))))
111	25, 110	sylbid 230	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘))))
112	111	adantr 481	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘))))
113		simpr 477	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → 𝑥 = (𝑝(ball‘𝐷)𝑟))
114	113	neeq1d 2853	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → (𝑥 ≠ ∅ ↔ (𝑝(ball‘𝐷)𝑟) ≠ ∅))
115		ral0 4076	. . . . . . . . . . . . . . . . . . 19 ⊢ ∀𝑘 ∈ ∅ (𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)
116		difeq2 3722	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝐼 → (𝐼 ∖ 𝑧) = (𝐼 ∖ 𝐼))
117		difid 3948	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐼 ∖ 𝐼) = ∅
118	116, 117	syl6eq 2672	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝐼 → (𝐼 ∖ 𝑧) = ∅)
119	118	raleqdv 3144	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝐼 → (∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘) ↔ ∀𝑘 ∈ ∅ (𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)))
120	119	rspcev 3309	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ Fin ∧ ∀𝑘 ∈ ∅ (𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘))
121	1, 115, 120	sylancl 694	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘))
122	121	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘))
123	122	biantrud 528	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → ((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ↔ ((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘))))
124		df-3an 1039	. . . . . . . . . . . . . . . 16 ⊢ ((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ↔ ((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)))
125	123, 124	syl6rbbr 279	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → ((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ↔ (𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘))))
126		eqeq1 2626	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑝(ball‘𝐷)𝑟) → (𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘) ↔ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))
127	125, 126	bi2anan9 917	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → (((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘)) ↔ ((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘))))
128	127	exbidv 1850	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → (∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘)) ↔ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘))))
129	112, 114, 128	3imtr4d 283	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → (𝑥 ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))))
130	129	ex 450	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → (𝑥 = (𝑝(ball‘𝐷)𝑟) → (𝑥 ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))))
131	130	rexlimdvva 3038	. . . . . . . . . 10 ⊢ (𝜑 → (∃𝑝 ∈ 𝐵 ∃𝑟 ∈ ℝ* 𝑥 = (𝑝(ball‘𝐷)𝑟) → (𝑥 ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))))
132	20, 131	sylbid 230	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ran (ball‘𝐷) → (𝑥 ≠ ∅ → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))))
133	132	impd 447	. . . . . . . 8 ⊢ (𝜑 → ((𝑥 ∈ ran (ball‘𝐷) ∧ 𝑥 ≠ ∅) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))))
134	13, 133	syl5bi 232	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (ran (ball‘𝐷) ∖ {∅}) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))))
135	134	alrimiv 1855	. . . . . 6 ⊢ (𝜑 → ∀𝑥(𝑥 ∈ (ran (ball‘𝐷) ∖ {∅}) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))))
136		ssab 3672	. . . . . 6 ⊢ ((ran (ball‘𝐷) ∖ {∅}) ⊆ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))} ↔ ∀𝑥(𝑥 ∈ (ran (ball‘𝐷) ∖ {∅}) → ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))))
137	135, 136	sylibr 224	. . . . 5 ⊢ (𝜑 → (ran (ball‘𝐷) ∖ {∅}) ⊆ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑔‘𝑘) ∈ ((TopOpen ∘ 𝑅)‘𝑘) ∧ ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) ∧ 𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘))})
138	137, 10	syl6sseqr 3652	. . . 4 ⊢ (𝜑 → (ran (ball‘𝐷) ∖ {∅}) ⊆ 𝐶)
139		ssv 3625	. . . . . . . . . 10 ⊢ ∞MetSp ⊆ V
140		fnssres 6004	. . . . . . . . . 10 ⊢ ((TopOpen Fn V ∧ ∞MetSp ⊆ V) → (TopOpen ↾ ∞MetSp) Fn ∞MetSp)
141	2, 139, 140	mp2an 708	. . . . . . . . 9 ⊢ (TopOpen ↾ ∞MetSp) Fn ∞MetSp
142		fvres 6207	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∞MetSp → ((TopOpen ↾ ∞MetSp)‘𝑥) = (TopOpen‘𝑥))
143		xmstps 22258	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∞MetSp → 𝑥 ∈ TopSp)
144		eqid 2622	. . . . . . . . . . . . 13 ⊢ (TopOpen‘𝑥) = (TopOpen‘𝑥)
145	144	tpstop 20741	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ TopSp → (TopOpen‘𝑥) ∈ Top)
146	143, 145	syl 17	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∞MetSp → (TopOpen‘𝑥) ∈ Top)
147	142, 146	eqeltrd 2701	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∞MetSp → ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top)
148	147	rgen 2922	. . . . . . . . 9 ⊢ ∀𝑥 ∈ ∞MetSp ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top
149		ffnfv 6388	. . . . . . . . 9 ⊢ ((TopOpen ↾ ∞MetSp):∞MetSp⟶Top ↔ ((TopOpen ↾ ∞MetSp) Fn ∞MetSp ∧ ∀𝑥 ∈ ∞MetSp ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top))
150	141, 148, 149	mpbir2an 955	. . . . . . . 8 ⊢ (TopOpen ↾ ∞MetSp):∞MetSp⟶Top
151		fco2 6059	. . . . . . . 8 ⊢ (((TopOpen ↾ ∞MetSp):∞MetSp⟶Top ∧ 𝑅:𝐼⟶∞MetSp) → (TopOpen ∘ 𝑅):𝐼⟶Top)
152	150, 3, 151	sylancr 695	. . . . . . 7 ⊢ (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
153		eqid 2622	. . . . . . . 8 ⊢ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) = X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)
154	10, 153	ptbasfi 21384	. . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ (TopOpen ∘ 𝑅):𝐼⟶Top) → 𝐶 = (fi‘({X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)))))
155	1, 152, 154	syl2anc 693	. . . . . 6 ⊢ (𝜑 → 𝐶 = (fi‘({X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)))))
156		eqid 2622	. . . . . . . . 9 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
157	156	mopntop 22245	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ Top)
158	18, 157	syl 17	. . . . . . 7 ⊢ (𝜑 → (MetOpen‘𝐷) ∈ Top)
159	14, 17, 15, 1, 5	prdsbas2 16129	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = X𝑘 ∈ 𝐼 (Base‘(𝑅‘𝑘)))
160	3, 74	sylan 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
161	3	ffvelrnda 6359	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ ∞MetSp)
162		xmstps 22258	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅‘𝑘) ∈ ∞MetSp → (𝑅‘𝑘) ∈ TopSp)
163	161, 162	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ TopSp)
164	36, 73	istps 20738	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅‘𝑘) ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑉))
165	163, 164	sylib 208	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐾 ∈ (TopOn‘𝑉))
166	160, 165	eqeltrd 2701	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) ∈ (TopOn‘𝑉))
167		toponuni 20719	. . . . . . . . . . . . . . 15 ⊢ (((TopOpen ∘ 𝑅)‘𝑘) ∈ (TopOn‘𝑉) → 𝑉 = ∪ ((TopOpen ∘ 𝑅)‘𝑘))
168	166, 167	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑉 = ∪ ((TopOpen ∘ 𝑅)‘𝑘))
169	36, 168	syl5eqr 2670	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (Base‘(𝑅‘𝑘)) = ∪ ((TopOpen ∘ 𝑅)‘𝑘))
170	169	ixpeq2dva 7923	. . . . . . . . . . . 12 ⊢ (𝜑 → X𝑘 ∈ 𝐼 (Base‘(𝑅‘𝑘)) = X𝑘 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑘))
171	159, 170	eqtrd 2656	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = X𝑘 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑘))
172		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑛))
173	172	unieqd 4446	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → ∪ ((TopOpen ∘ 𝑅)‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑛))
174	173	cbvixpv 7926	. . . . . . . . . . 11 ⊢ X𝑘 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑘) = X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)
175	171, 174	syl6eq 2672	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛))
176	156	mopntopon 22244	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ (TopOn‘𝐵))
177	18, 176	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (MetOpen‘𝐷) ∈ (TopOn‘𝐵))
178		toponmax 20730	. . . . . . . . . . 11 ⊢ ((MetOpen‘𝐷) ∈ (TopOn‘𝐵) → 𝐵 ∈ (MetOpen‘𝐷))
179	177, 178	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (MetOpen‘𝐷))
180	175, 179	eqeltrrd 2702	. . . . . . . . 9 ⊢ (𝜑 → X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ∈ (MetOpen‘𝐷))
181	180	snssd 4340	. . . . . . . 8 ⊢ (𝜑 → {X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)} ⊆ (MetOpen‘𝐷))
182	175	mpteq1d 4738	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) = (𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)))
183	182	ad2antrr 762	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) = (𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)))
184	183	cnveqd 5298	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → ◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) = ◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)))
185	184	imaeq1d 5465	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢))
186		fveq1 6190	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑝 → (𝑤‘𝑘) = (𝑝‘𝑘))
187	186	eleq1d 2686	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑝 → ((𝑤‘𝑘) ∈ 𝑢 ↔ (𝑝‘𝑘) ∈ 𝑢))
188		eqid 2622	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘))
189	188	mptpreima 5628	. . . . . . . . . . . . . . . . . . 19 ⊢ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑤 ∈ 𝐵 ∣ (𝑤‘𝑘) ∈ 𝑢}
190	187, 189	elrab2 3366	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))
191	161, 38	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
192	191	adantr 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝐸 ∈ (∞Met‘𝑉))
193		simprl 794	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝑢 ∈ 𝐾)
194	161, 76	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐾 = (MetOpen‘𝐸))
195	194	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝐾 = (MetOpen‘𝐸))
196	193, 195	eleqtrd 2703	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝑢 ∈ (MetOpen‘𝐸))
197		simprrr 805	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → (𝑝‘𝑘) ∈ 𝑢)
198	55	mopni2 22298	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑢 ∈ (MetOpen‘𝐸) ∧ (𝑝‘𝑘) ∈ 𝑢) → ∃𝑟 ∈ ℝ+ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)
199	192, 196, 197, 198	syl3anc 1326	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → ∃𝑟 ∈ ℝ+ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)
200	18	ad3antrrr 766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝐷 ∈ (∞Met‘𝐵))
201		simprrl 804	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝑝 ∈ 𝐵)
202	201	adantr 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑝 ∈ 𝐵)
203		rpxr 11840	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
204	203	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ*)
205	156	blopn 22305	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) → (𝑝(ball‘𝐷)𝑟) ∈ (MetOpen‘𝐷))
206	200, 202, 204, 205	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ∈ (MetOpen‘𝐷))
207		simprl 794	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ+)
208		blcntr 22218	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+) → 𝑝 ∈ (𝑝(ball‘𝐷)𝑟))
209	200, 202, 207, 208	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑝 ∈ (𝑝(ball‘𝐷)𝑟))
210		blssm 22223	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) → (𝑝(ball‘𝐷)𝑟) ⊆ 𝐵)
211	200, 202, 204, 210	syl3anc 1326	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ 𝐵)
212		simplrr 801	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)
213		simplll 798	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝜑)
214		rpgt0 11844	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑟 ∈ ℝ+ → 0 < 𝑟)
215	214	ad2antrl 764	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 0 < 𝑟)
216	213, 202, 204, 215, 97	syl121anc 1331	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟))
217	216	eleq2d 2687	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑤 ∈ (𝑝(ball‘𝐷)𝑟) ↔ 𝑤 ∈ X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟)))
218	217	biimpa 501	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → 𝑤 ∈ X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟))
219		vex 3203	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑤 ∈ V
220	219	elixp 7915	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑤 ∈ X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟) ↔ (𝑤 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟)))
221	220	simprbi 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑤 ∈ X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟) → ∀𝑘 ∈ 𝐼 (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟))
222	218, 221	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → ∀𝑘 ∈ 𝐼 (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟))
223		simp-4r 807	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → 𝑘 ∈ 𝐼)
224		rsp 2929	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∀𝑘 ∈ 𝐼 (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟) → (𝑘 ∈ 𝐼 → (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟)))
225	222, 223, 224	sylc 65	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟))
226	212, 225	sseldd 3604	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) ∧ 𝑤 ∈ (𝑝(ball‘𝐷)𝑟)) → (𝑤‘𝑘) ∈ 𝑢)
227	211, 226	ssrabdv 3681	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ {𝑤 ∈ 𝐵 ∣ (𝑤‘𝑘) ∈ 𝑢})
228	227, 189	syl6sseqr 3652	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))
229		eleq2 2690	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = (𝑝(ball‘𝐷)𝑟) → (𝑝 ∈ 𝑦 ↔ 𝑝 ∈ (𝑝(ball‘𝐷)𝑟)))
230		sseq1 3626	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = (𝑝(ball‘𝐷)𝑟) → (𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑝(ball‘𝐷)𝑟) ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)))
231	229, 230	anbi12d 747	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 = (𝑝(ball‘𝐷)𝑟) → ((𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)) ↔ (𝑝 ∈ (𝑝(ball‘𝐷)𝑟) ∧ (𝑝(ball‘𝐷)𝑟) ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))))
232	231	rspcev 3309	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑝(ball‘𝐷)𝑟) ∈ (MetOpen‘𝐷) ∧ (𝑝 ∈ (𝑝(ball‘𝐷)𝑟) ∧ (𝑝(ball‘𝐷)𝑟) ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)))
233	206, 209, 228, 232	syl12anc 1324	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)))
234	199, 233	rexlimddv 3035	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)))
235	234	expr 643	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → ((𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))))
236	190, 235	syl5bi 232	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (𝑝 ∈ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))))
237	236	ralrimiv 2965	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → ∀𝑝 ∈ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)))
238	158	ad2antrr 762	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (MetOpen‘𝐷) ∈ Top)
239		eltop2 20779	. . . . . . . . . . . . . . . . 17 ⊢ ((MetOpen‘𝐷) ∈ Top → ((◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑝 ∈ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))))
240	238, 239	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → ((◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑝 ∈ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))))
241	237, 240	mpbird 247	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
242	185, 241	eqeltrrd 2702	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
243	242	ralrimiva 2966	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∀𝑢 ∈ 𝐾 (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
244	160	raleqdv 3144	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑢 ∈ 𝐾 (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷)))
245	243, 244	mpbird 247	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
246	245	ralrimiva 2966	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝐼 ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
247		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑚))
248		fveq2 6191	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑚 → (𝑤‘𝑘) = (𝑤‘𝑚))
249	248	mpteq2dv 4745	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) = (𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)))
250	249	cnveqd 5298	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → ◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) = ◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)))
251	250	imaeq1d 5465	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢))
252	251	eleq1d 2686	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ((◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷)))
253	247, 252	raleqbidv 3152	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑚 → (∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷)))
254	253	cbvralv 3171	. . . . . . . . . . 11 ⊢ (∀𝑘 ∈ 𝐼 ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ ∀𝑚 ∈ 𝐼 ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷))
255	246, 254	sylib 208	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ 𝐼 ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷))
256		eqid 2622	. . . . . . . . . . 11 ⊢ (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)) = (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢))
257	256	fmpt2x 7236	. . . . . . . . . 10 ⊢ (∀𝑚 ∈ 𝐼 ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷) ↔ (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)):∪ 𝑚 ∈ 𝐼 ({𝑚} × ((TopOpen ∘ 𝑅)‘𝑚))⟶(MetOpen‘𝐷))
258	255, 257	sylib 208	. . . . . . . . 9 ⊢ (𝜑 → (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)):∪ 𝑚 ∈ 𝐼 ({𝑚} × ((TopOpen ∘ 𝑅)‘𝑚))⟶(MetOpen‘𝐷))
259		frn 6053	. . . . . . . . 9 ⊢ ((𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)):∪ 𝑚 ∈ 𝐼 ({𝑚} × ((TopOpen ∘ 𝑅)‘𝑚))⟶(MetOpen‘𝐷) → ran (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)) ⊆ (MetOpen‘𝐷))
260	258, 259	syl 17	. . . . . . . 8 ⊢ (𝜑 → ran (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)) ⊆ (MetOpen‘𝐷))
261	181, 260	unssd 3789	. . . . . . 7 ⊢ (𝜑 → ({X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢))) ⊆ (MetOpen‘𝐷))
262		fiss 8330	. . . . . . 7 ⊢ (((MetOpen‘𝐷) ∈ Top ∧ ({X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢))) ⊆ (MetOpen‘𝐷)) → (fi‘({X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)))) ⊆ (fi‘(MetOpen‘𝐷)))
263	158, 261, 262	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (fi‘({X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)} ∪ ran (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)))) ⊆ (fi‘(MetOpen‘𝐷)))
264	155, 263	eqsstrd 3639	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (fi‘(MetOpen‘𝐷)))
265		fitop 20705	. . . . . . 7 ⊢ ((MetOpen‘𝐷) ∈ Top → (fi‘(MetOpen‘𝐷)) = (MetOpen‘𝐷))
266	158, 265	syl 17	. . . . . 6 ⊢ (𝜑 → (fi‘(MetOpen‘𝐷)) = (MetOpen‘𝐷))
267	156	mopnval 22243	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
268	18, 267	syl 17	. . . . . . 7 ⊢ (𝜑 → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
269		tgdif0 20796	. . . . . . 7 ⊢ (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘ran (ball‘𝐷))
270	268, 269	syl6eqr 2674	. . . . . 6 ⊢ (𝜑 → (MetOpen‘𝐷) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
271	266, 270	eqtrd 2656	. . . . 5 ⊢ (𝜑 → (fi‘(MetOpen‘𝐷)) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
272	264, 271	sseqtrd 3641	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ (topGen‘(ran (ball‘𝐷) ∖ {∅})))
273		2basgen 20794	. . . 4 ⊢ (((ran (ball‘𝐷) ∖ {∅}) ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘(ran (ball‘𝐷) ∖ {∅}))) → (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘𝐶))
274	138, 272, 273	syl2anc 693	. . 3 ⊢ (𝜑 → (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘𝐶))
275	12, 274	eqtr4d 2659	. 2 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
276		prdsxms.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝑌)
277	14, 15, 1, 5, 276	prdstopn 21431	. 2 ⊢ (𝜑 → 𝐽 = (∏t‘(TopOpen ∘ 𝑅)))
278	275, 277, 270	3eqtr4d 2666	1 ⊢ (𝜑 → 𝐽 = (MetOpen‘𝐷))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  {csn 4177  ∪ cuni 4436  ∪ ciun 4520   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  ^◡ccnv 5113  ran crn 5115   ↾ cres 5116   “ cima 5117   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650   ↦ cmpt2 6652  Xcixp 7908  Fincfn 7955  ficfi 8316  0cc0 9936  ℝ^*cxr 10073   < clt 10074  ℝ⁺crp 11832  Basecbs 15857  distcds 15950  TopOpenctopn 16082  topGenctg 16098  ∏_tcpt 16099  X_scprds 16106  ∞Metcxmt 19731  ballcbl 19733  MetOpencmopn 19736  Topctop 20698  TopOnctopon 20715  TopSpctps 20736  ∞MetSpcxme 22122

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fi 8317  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-icc 12182  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-topgen 16104  df-pt 16105  df-prds 16108  df-psmet 19738  df-xmet 19739  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-xms 22125

This theorem is referenced by:  prdsxms  22335