Theorem prdsxmslem2 24739

Description: Lemma for prdsxms 24740. 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 17502	. . . . 5 ⊢ TopOpen Fn V
3		prdsxms.r	. . . . . . 7 ⊢ (𝜑 → 𝑅:𝐼⟶∞MetSp)
4	3	ffnd 6710	. . . . . 6 ⊢ (𝜑 → 𝑅 Fn 𝐼)
5		dffn2 6711	. . . . . 6 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅:𝐼⟶V)
6	4, 5	sylib 221	. . . . 5 ⊢ (𝜑 → 𝑅:𝐼⟶V)
7		fnfco 6747	. . . . 5 ⊢ ((TopOpen Fn V ∧ 𝑅:𝐼⟶V) → (TopOpen ∘ 𝑅) Fn 𝐼)
8	2, 6, 7	sylancr 599	. . . 4 ⊢ (𝜑 → (TopOpen ∘ 𝑅) Fn 𝐼)
9		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))}
10	9	ptval 23780	. . . 4 ⊢ ((𝐼 ∈ Fin ∧ (TopOpen ∘ 𝑅) Fn 𝐼) → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘𝐶))
11	1, 8, 10	syl2anc 596	. . 3 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘𝐶))
12		eldifsn 4755	. . . . . . . 8 ⊢ (𝑥 ∈ (ran (ball‘𝐷) ∖ {∅}) ↔ (𝑥 ∈ ran (ball‘𝐷) ∧ 𝑥 ≠ ∅))
13		prdsxms.y	. . . . . . . . . . . 12 ⊢ 𝑌 = (𝑆Xs𝑅)
14		prdsxms.s	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
15		prdsxms.d	. . . . . . . . . . . 12 ⊢ 𝐷 = (dist‘𝑌)
16		prdsxms.b	. . . . . . . . . . . 12 ⊢ 𝐵 = (Base‘𝑌)
17	13, 14, 1, 15, 16, 3	prdsxmslem1 24738	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))
18		blrn 24619	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (∞Met‘𝐵) → (𝑥 ∈ ran (ball‘𝐷) ↔ ∃𝑝 ∈ 𝐵 ∃𝑟 ∈ ℝ* 𝑥 = (𝑝(ball‘𝐷)𝑟)))
19	17, 18	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ ran (ball‘𝐷) ↔ ∃𝑝 ∈ 𝐵 ∃𝑟 ∈ ℝ* 𝑥 = (𝑝(ball‘𝐷)𝑟)))
20	17	adantr 486	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → 𝐷 ∈ (∞Met‘𝐵))
21		simprl 783	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → 𝑝 ∈ 𝐵)
22		simprr 785	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → 𝑟 ∈ ℝ*)
23		xbln0 24624	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
24	20, 21, 22, 23	syl3anc 1398	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → ((𝑝(ball‘𝐷)𝑟) ≠ ∅ ↔ 0 < 𝑟))
25	1	3ad2ant1 1151	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐼 ∈ Fin)
26	25	mptexd 7229	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟)) ∈ V)
27		ovex 7452	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟) ∈ V
28	27	rgenw 3085	. . . . . . . . . . . . . . . . . 18 ⊢ ∀𝑛 ∈ 𝐼 ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟) ∈ V
29		eqid 2765	. . . . . . . . . . . . . . . . . . 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))
30	29	fnmpt 6679	. . . . . . . . . . . . . . . . . 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)
31	28, 30	mp1i 14	. . . . . . . . . . . . . . . . 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)
32	3	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅:𝐼⟶∞MetSp)
33	32	ffvelcdmda 7083	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ ∞MetSp)
34		prdsxms.v	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝑉 = (Base‘(𝑅‘𝑘))
35		prdsxms.e	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐸 = ((dist‘(𝑅‘𝑘)) ↾ (𝑉 × 𝑉))
36	34, 35	xmsxmet 24666	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅‘𝑘) ∈ ∞MetSp → 𝐸 ∈ (∞Met‘𝑉))
37	33, 36	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
38		eqid 2765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))) = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))
39		eqid 2765	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))) = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))
40	14	3ad2ant1 1151	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑆 ∈ 𝑊)
41	33	ralrimiva 3159	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∀𝑘 ∈ 𝐼 (𝑅‘𝑘) ∈ ∞MetSp)
42		simp2l 1218	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝 ∈ 𝐵)
43	32	feqmptd 6953	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅 = (𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))
44	43	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑆Xs𝑅) = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))
45	13, 44	eqtrid 2812	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑌 = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))))
46	45	fveq2d 6889	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (Base‘𝑌) = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))))
47	16, 46	eqtrid 2812	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐵 = (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))))
48	42, 47	eleqtrd 2867	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝 ∈ (Base‘(𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))))
49	38, 39, 40, 25, 41, 34, 48	prdsbascl 17560	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → ∀𝑘 ∈ 𝐼 (𝑝‘𝑘) ∈ 𝑉)
50	49	r19.21bi 3259	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → (𝑝‘𝑘) ∈ 𝑉)
51		simp2r 1219	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑟 ∈ ℝ*)
52	51	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → 𝑟 ∈ ℝ*)
53		eqid 2765	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (MetOpen‘𝐸) = (MetOpen‘𝐸)
54	53	blopn 24710	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ (𝑝‘𝑘) ∈ 𝑉 ∧ 𝑟 ∈ ℝ*) → ((𝑝‘𝑘)(ball‘𝐸)𝑟) ∈ (MetOpen‘𝐸))
55	37, 50, 52, 54	syl3anc 1398	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → ((𝑝‘𝑘)(ball‘𝐸)𝑟) ∈ (MetOpen‘𝐸))
56		2fveq3 6890	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑘 → (dist‘(𝑅‘𝑛)) = (dist‘(𝑅‘𝑘)))
57		2fveq3 6890	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑛 = 𝑘 → (Base‘(𝑅‘𝑛)) = (Base‘(𝑅‘𝑘)))
58	57, 34	eqtr4di 2818	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑛 = 𝑘 → (Base‘(𝑅‘𝑛)) = 𝑉)
59	58	sqxpeqd 5695	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑛 = 𝑘 → ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛))) = (𝑉 × 𝑉))
60	56, 59	reseq12d 5981	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 = 𝑘 → ((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))) = ((dist‘(𝑅‘𝑘)) ↾ (𝑉 × 𝑉)))
61	60, 35	eqtr4di 2818	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑛 = 𝑘 → ((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))) = 𝐸)
62	61	fveq2d 6889	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑘 → (ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛))))) = (ball‘𝐸))
63		fveq2 6885	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑘 → (𝑝‘𝑛) = (𝑝‘𝑘))
64		eqidd 2766	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑘 → 𝑟 = 𝑟)
65	62, 63, 64	oveq123d 7440	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑘 → ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟) = ((𝑝‘𝑘)(ball‘𝐸)𝑟))
66		ovex 7452	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ∈ V
67	65, 29, 66	fvmpt 6993	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 ∈ 𝐼 → ((𝑛 ∈ 𝐼 ↦ ((𝑝‘𝑛)(ball‘((dist‘(𝑅‘𝑛)) ↾ ((Base‘(𝑅‘𝑛)) × (Base‘(𝑅‘𝑛)))))𝑟))‘𝑘) = ((𝑝‘𝑘)(ball‘𝐸)𝑟))
68	67	adantl 487	. . . . . . . . . . . . . . . . . . 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))
69		fvco3 6985	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑅:𝐼⟶∞MetSp ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (TopOpen‘(𝑅‘𝑘)))
70		prdsxms.k	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ 𝐾 = (TopOpen‘(𝑅‘𝑘))
71	69, 70	eqtr4di 2818	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅:𝐼⟶∞MetSp ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
72	32, 71	sylan 592	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
73	70, 34, 35	xmstopn 24661	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑅‘𝑘) ∈ ∞MetSp → 𝐾 = (MetOpen‘𝐸))
74	33, 73	syl 18	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → 𝐾 = (MetOpen‘𝐸))
75	72, 74	eqtrd 2800	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = (MetOpen‘𝐸))
76	55, 68, 75	3eltr4d 2880	. . . . . . . . . . . . . . . . . 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))
77	76	ralrimiva 3159	. . . . . . . . . . . . . . . . 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))
78	32	feqmptd 6953	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑅 = (𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))
79	78	oveq2d 7435	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑆Xs𝑅) = (𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))
80	13, 79	eqtrid 2812	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑌 = (𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))
81	80	fveq2d 6889	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (dist‘𝑌) = (dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))
82	15, 81	eqtrid 2812	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐷 = (dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))
83	82	fveq2d 6889	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (ball‘𝐷) = (ball‘(dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))))
84	83	oveqd 7436	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘𝐷)𝑟) = (𝑝(ball‘(dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))𝑟))
85		fveq2 6885	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑘 → (𝑅‘𝑛) = (𝑅‘𝑘))
86	85	cbvmptv 5217	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)) = (𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘))
87	86	oveq2i 7430	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))) = (𝑆Xs(𝑘 ∈ 𝐼 ↦ (𝑅‘𝑘)))
88		eqid 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))) = (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))
89		eqid 2765	. . . . . . . . . . . . . . . . . . 19 ⊢ (dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))) = (dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛))))
90	80	fveq2d 6889	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (Base‘𝑌) = (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))
91	16, 90	eqtrid 2812	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝐵 = (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))
92	42, 91	eleqtrd 2867	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 𝑝 ∈ (Base‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))
93		simp3 1156	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → 0 < 𝑟)
94	87, 88, 34, 35, 89, 40, 25, 33, 37, 92, 51, 93	prdsbl 24701	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘(dist‘(𝑆Xs(𝑛 ∈ 𝐼 ↦ (𝑅‘𝑛)))))𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟))
95	84, 94	eqtrd 2800	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) ∧ 0 < 𝑟) → (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟))
96		fneq1 6630	. . . . . . . . . . . . . . . . . . . . 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))
97		fveq1 6884	. . . . . . . . . . . . . . . . . . . . . . 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))
98	97	eleq1d 2850	. . . . . . . . . . . . . . . . . . . . . 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)))
99	98	ralbidv 3190	. . . . . . . . . . . . . . . . . . . . 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)))
100	96, 99	anbi12d 644	. . . . . . . . . . . . . . . . . . . 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))))
101	97, 67	sylan9eq 2820	. . . . . . . . . . . . . . . . . . . . . 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))
102	101	ixpeq2dva 8916	. . . . . . . . . . . . . . . . . . . . 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))
103	102	eqeq2d 2776	. . . . . . . . . . . . . . . . . . . 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)))
104	100, 103	anbi12d 644	. . . . . . . . . . . . . . . . . . 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))))
105	104	spcegv 3558	. . . . . . . . . . . . . . . . . 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))))
106	105	3impib 1134	. . . . . . . . . . . . . . . . 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)))
107	26, 31, 77, 95, 106	syl121anc 1402	. . . . . . . . . . . . . . . 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)))
108	107	3expia 1139	. . . . . . . . . . . . . . 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))))
109	24, 108	sylbid 243	. . . . . . . . . . . . . 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))))
110	109	adantr 486	. . . . . . . . . . . . 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))))
111		simpr 490	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → 𝑥 = (𝑝(ball‘𝐷)𝑟))
112	111	neeq1d 3019	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) ∧ 𝑥 = (𝑝(ball‘𝐷)𝑟)) → (𝑥 ≠ ∅ ↔ (𝑝(ball‘𝐷)𝑟) ≠ ∅))
113		df-3an 1105	. . . . . . . . . . . . . . . 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)))
114		ral0 4461	. . . . . . . . . . . . . . . . . . 19 ⊢ ∀𝑘 ∈ ∅ (𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)
115		difeq2 4075	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑧 = 𝐼 → (𝐼 ∖ 𝑧) = (𝐼 ∖ 𝐼))
116		difid 4332	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝐼 ∖ 𝐼) = ∅
117	115, 116	eqtrdi 2816	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑧 = 𝐼 → (𝐼 ∖ 𝑧) = ∅)
118	117	raleqdv 3325	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑧 = 𝐼 → (∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘) ↔ ∀𝑘 ∈ ∅ (𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)))
119	118	rspcev 3583	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐼 ∈ Fin ∧ ∀𝑘 ∈ ∅ (𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘)) → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘))
120	1, 114, 119	sylancl 598	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘))
121	120	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*)) → ∃𝑧 ∈ Fin ∀𝑘 ∈ (𝐼 ∖ 𝑧)(𝑔‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑘))
122	121	biantrud 541	. . . . . . . . . . . . . . . 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))))
123	113, 122	bitr4id 293	. . . . . . . . . . . . . . 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))))
124		eqeq1 2769	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = (𝑝(ball‘𝐷)𝑟) → (𝑥 = X𝑘 ∈ 𝐼 (𝑔‘𝑘) ↔ (𝑝(ball‘𝐷)𝑟) = X𝑘 ∈ 𝐼 (𝑔‘𝑘)))
125	123, 124	bi2anan9 650	. . . . . . . . . . . . . 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))))
126	125	exbidv 1954	. . . . . . . . . . . . 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))))
127	110, 112, 126	3imtr4d 297	. . . . . . . . . . . 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))))
128	127	ex 418	. . . . . . . . . . 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)))))
129	128	rexlimdvva 3224	. . . . . . . . . 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)))))
130	19, 129	sylbid 243	. . . . . . . . 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)))))
131	130	impd 416	. . . . . . . 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))))
132	12, 131	biimtrid 245	. . . . . . 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))))
133	132	alrimiv 1960	. . . . . 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))))
134		ssab 4018	. . . . . 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))))
135	133, 134	sylibr 237	. . . . 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))})
136	135, 9	sseqtrrdi 3979	. . . 4 ⊢ (𝜑 → (ran (ball‘𝐷) ∖ {∅}) ⊆ 𝐶)
137		ssv 3962	. . . . . . . . . 10 ⊢ ∞MetSp ⊆ V
138		fnssres 6662	. . . . . . . . . 10 ⊢ ((TopOpen Fn V ∧ ∞MetSp ⊆ V) → (TopOpen ↾ ∞MetSp) Fn ∞MetSp)
139	2, 137, 138	mp2an 705	. . . . . . . . 9 ⊢ (TopOpen ↾ ∞MetSp) Fn ∞MetSp
140		fvres 6904	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∞MetSp → ((TopOpen ↾ ∞MetSp)‘𝑥) = (TopOpen‘𝑥))
141		xmstps 24663	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∞MetSp → 𝑥 ∈ TopSp)
142		eqid 2765	. . . . . . . . . . . . 13 ⊢ (TopOpen‘𝑥) = (TopOpen‘𝑥)
143	142	tpstop 23146	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ TopSp → (TopOpen‘𝑥) ∈ Top)
144	141, 143	syl 18	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∞MetSp → (TopOpen‘𝑥) ∈ Top)
145	140, 144	eqeltrd 2865	. . . . . . . . . 10 ⊢ (𝑥 ∈ ∞MetSp → ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top)
146	145	rgen 3083	. . . . . . . . 9 ⊢ ∀𝑥 ∈ ∞MetSp ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top
147		ffnfv 7118	. . . . . . . . 9 ⊢ ((TopOpen ↾ ∞MetSp):∞MetSp⟶Top ↔ ((TopOpen ↾ ∞MetSp) Fn ∞MetSp ∧ ∀𝑥 ∈ ∞MetSp ((TopOpen ↾ ∞MetSp)‘𝑥) ∈ Top))
148	139, 146, 147	mpbir2an 724	. . . . . . . 8 ⊢ (TopOpen ↾ ∞MetSp):∞MetSp⟶Top
149		fco2 6736	. . . . . . . 8 ⊢ (((TopOpen ↾ ∞MetSp):∞MetSp⟶Top ∧ 𝑅:𝐼⟶∞MetSp) → (TopOpen ∘ 𝑅):𝐼⟶Top)
150	148, 3, 149	sylancr 599	. . . . . . 7 ⊢ (𝜑 → (TopOpen ∘ 𝑅):𝐼⟶Top)
151		eqid 2765	. . . . . . . 8 ⊢ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) = X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)
152	9, 151	ptbasfi 23791	. . . . . . 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)))))
153	1, 150, 152	syl2anc 596	. . . . . 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)))))
154		eqid 2765	. . . . . . . . 9 ⊢ (MetOpen‘𝐷) = (MetOpen‘𝐷)
155	154	mopntop 24650	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ Top)
156	17, 155	syl 18	. . . . . . 7 ⊢ (𝜑 → (MetOpen‘𝐷) ∈ Top)
157	13, 16, 14, 1, 4	prdsbas2 17546	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 = X𝑘 ∈ 𝐼 (Base‘(𝑅‘𝑘)))
158	3, 71	sylan 592	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) = 𝐾)
159	3	ffvelcdmda 7083	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ ∞MetSp)
160		xmstps 24663	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑅‘𝑘) ∈ ∞MetSp → (𝑅‘𝑘) ∈ TopSp)
161	159, 160	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑅‘𝑘) ∈ TopSp)
162	34, 70	istps 23143	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑅‘𝑘) ∈ TopSp ↔ 𝐾 ∈ (TopOn‘𝑉))
163	161, 162	sylib 221	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐾 ∈ (TopOn‘𝑉))
164	158, 163	eqeltrd 2865	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ((TopOpen ∘ 𝑅)‘𝑘) ∈ (TopOn‘𝑉))
165		toponuni 23123	. . . . . . . . . . . . . . 15 ⊢ (((TopOpen ∘ 𝑅)‘𝑘) ∈ (TopOn‘𝑉) → 𝑉 = ∪ ((TopOpen ∘ 𝑅)‘𝑘))
166	164, 165	syl 18	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝑉 = ∪ ((TopOpen ∘ 𝑅)‘𝑘))
167	34, 166	eqtr3id 2814	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (Base‘(𝑅‘𝑘)) = ∪ ((TopOpen ∘ 𝑅)‘𝑘))
168	167	ixpeq2dva 8916	. . . . . . . . . . . 12 ⊢ (𝜑 → X𝑘 ∈ 𝐼 (Base‘(𝑅‘𝑘)) = X𝑘 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑘))
169	157, 168	eqtrd 2800	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 = X𝑘 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑘))
170		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑛))
171	170	unieqd 4887	. . . . . . . . . . . 12 ⊢ (𝑘 = 𝑛 → ∪ ((TopOpen ∘ 𝑅)‘𝑘) = ∪ ((TopOpen ∘ 𝑅)‘𝑛))
172	171	cbvixpv 8919	. . . . . . . . . . 11 ⊢ X𝑘 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑘) = X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)
173	169, 172	eqtrdi 2816	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 = X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛))
174	154	mopntopon 24649	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) ∈ (TopOn‘𝐵))
175	17, 174	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (MetOpen‘𝐷) ∈ (TopOn‘𝐵))
176		toponmax 23135	. . . . . . . . . . 11 ⊢ ((MetOpen‘𝐷) ∈ (TopOn‘𝐵) → 𝐵 ∈ (MetOpen‘𝐷))
177	175, 176	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ (MetOpen‘𝐷))
178	173, 177	eqeltrrd 2866	. . . . . . . . 9 ⊢ (𝜑 → X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ∈ (MetOpen‘𝐷))
179	178	snssd 4754	. . . . . . . 8 ⊢ (𝜑 → {X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛)} ⊆ (MetOpen‘𝐷))
180	173	mpteq1d 5203	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) = (𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)))
181	180	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) = (𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)))
182	181	cnveqd 5863	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → ◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) = ◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)))
183	182	imaeq1d 6063	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢))
184		fveq1 6884	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 = 𝑝 → (𝑤‘𝑘) = (𝑝‘𝑘))
185	184	eleq1d 2850	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑤 = 𝑝 → ((𝑤‘𝑘) ∈ 𝑢 ↔ (𝑝‘𝑘) ∈ 𝑢))
186		eqid 2765	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘))
187	186	mptpreima 6241	. . . . . . . . . . . . . . . . . . 19 ⊢ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) = {𝑤 ∈ 𝐵 ∣ (𝑤‘𝑘) ∈ 𝑢}
188	185, 187	elrab2 3656	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∈ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))
189	159, 36	syl 18	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
190	189	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝐸 ∈ (∞Met‘𝑉))
191		simprl 783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝑢 ∈ 𝐾)
192	159, 73	syl 18	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐾 = (MetOpen‘𝐸))
193	192	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝐾 = (MetOpen‘𝐸))
194	191, 193	eleqtrd 2867	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝑢 ∈ (MetOpen‘𝐸))
195		simprrr 794	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → (𝑝‘𝑘) ∈ 𝑢)
196	53	mopni2 24703	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐸 ∈ (∞Met‘𝑉) ∧ 𝑢 ∈ (MetOpen‘𝐸) ∧ (𝑝‘𝑘) ∈ 𝑢) → ∃𝑟 ∈ ℝ+ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)
197	190, 194, 195, 196	syl3anc 1398	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → ∃𝑟 ∈ ℝ+ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)
198	17	ad3antrrr 743	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝐷 ∈ (∞Met‘𝐵))
199		simprrl 793	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → 𝑝 ∈ 𝐵)
200	199	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑝 ∈ 𝐵)
201		rpxr 13044	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
202	201	ad2antrl 741	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ*)
203	154	blopn 24710	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) → (𝑝(ball‘𝐷)𝑟) ∈ (MetOpen‘𝐷))
204	198, 200, 202, 203	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . 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))
205		simprl 783	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑟 ∈ ℝ+)
206		blcntr 24623	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+) → 𝑝 ∈ (𝑝(ball‘𝐷)𝑟))
207	198, 200, 205, 206	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝑝 ∈ (𝑝(ball‘𝐷)𝑟))
208		blssm 24628	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑝 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) → (𝑝(ball‘𝐷)𝑟) ⊆ 𝐵)
209	198, 200, 202, 208	syl3anc 1398	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → (𝑝(ball‘𝐷)𝑟) ⊆ 𝐵)
210		simplrr 790	. . . . . . . . . . . . . . . . . . . . . . . 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)
211		simplll 787	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 𝜑)
212		rpgt0 13047	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑟 ∈ ℝ+ → 0 < 𝑟)
213	212	ad2antrl 741	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) ∧ (𝑟 ∈ ℝ+ ∧ ((𝑝‘𝑘)(ball‘𝐸)𝑟) ⊆ 𝑢)) → 0 < 𝑟)
214	211, 200, 202, 213, 95	syl121anc 1402	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
215	214	eleq2d 2851	. . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
216	215	biimpa 482	. . . . . . . . . . . . . . . . . . . . . . . . . 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))
217		vex 3461	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑤 ∈ V
218	217	elixp 8908	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑤 ∈ X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟) ↔ (𝑤 Fn 𝐼 ∧ ∀𝑘 ∈ 𝐼 (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟)))
219	218	simprbi 503	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑤 ∈ X𝑘 ∈ 𝐼 ((𝑝‘𝑘)(ball‘𝐸)𝑟) → ∀𝑘 ∈ 𝐼 (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟))
220	216, 219	syl 18	. . . . . . . . . . . . . . . . . . . . . . . . 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))
221		simp-4r 796	. . . . . . . . . . . . . . . . . . . . . . . . 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)
222		rsp 3255	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∀𝑘 ∈ 𝐼 (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟) → (𝑘 ∈ 𝐼 → (𝑤‘𝑘) ∈ ((𝑝‘𝑘)(ball‘𝐸)𝑟)))
223	220, 221, 222	sylc 66	. . . . . . . . . . . . . . . . . . . . . . . 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))
224	210, 223	sseldd 3939	. . . . . . . . . . . . . . . . . . . . . . 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)
225	209, 224	ssrabdv 4028	. . . . . . . . . . . . . . . . . . . . . 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})
226	225, 187	sseqtrrdi 3979	. . . . . . . . . . . . . . . . . . . . 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))
227		eleq2 2854	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = (𝑝(ball‘𝐷)𝑟) → (𝑝 ∈ 𝑦 ↔ 𝑝 ∈ (𝑝(ball‘𝐷)𝑟)))
228		sseq1 3963	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 = (𝑝(ball‘𝐷)𝑟) → (𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) ↔ (𝑝(ball‘𝐷)𝑟) ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)))
229	227, 228	anbi12d 644	. . . . . . . . . . . . . . . . . . . . . 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))))
230	229	rspcev 3583	. . . . . . . . . . . . . . . . . . . . 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)))
231	204, 207, 226, 230	syl12anc 850	. . . . . . . . . . . . . . . . . . . 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)))
232	197, 231	rexlimddv 3174	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ (𝑢 ∈ 𝐾 ∧ (𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢))) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)))
233	232	expr 462	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → ((𝑝 ∈ 𝐵 ∧ (𝑝‘𝑘) ∈ 𝑢) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))))
234	188, 233	biimtrid 245	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (𝑝 ∈ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) → ∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢))))
235	234	ralrimiv 3158	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → ∀𝑝 ∈ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)∃𝑦 ∈ (MetOpen‘𝐷)(𝑝 ∈ 𝑦 ∧ 𝑦 ⊆ (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢)))
236	156	ad2antrr 739	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (MetOpen‘𝐷) ∈ Top)
237		eltop2 23184	. . . . . . . . . . . . . . . . 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))))
238	236, 237	syl 18	. . . . . . . . . . . . . . . 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))))
239	235, 238	mpbird 260	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (◡(𝑤 ∈ 𝐵 ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
240	183, 239	eqeltrrd 2866	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑢 ∈ 𝐾) → (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
241	240	ralrimiva 3159	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∀𝑢 ∈ 𝐾 (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
242	241, 158	raleqtrrdv 3329	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
243	242	ralrimiva 3159	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑘 ∈ 𝐼 ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑘)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) ∈ (MetOpen‘𝐷))
244		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑚 → ((TopOpen ∘ 𝑅)‘𝑘) = ((TopOpen ∘ 𝑅)‘𝑚))
245		fveq2 6885	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑚 → (𝑤‘𝑘) = (𝑤‘𝑚))
246	245	mpteq2dv 5207	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑚 → (𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) = (𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)))
247	246	cnveqd 5863	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑚 → ◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) = ◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)))
248	247	imaeq1d 6063	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑚 → (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢))
249	248	eleq1d 2850	. . . . . . . . . . . . 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)))
250	244, 249	raleqbidv 3340	. . . . . . . . . . . 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)))
251	250	cbvralvw 3245	. . . . . . . . . . 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))
252	243, 251	sylib 221	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑚 ∈ 𝐼 ∀𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚)(◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢) ∈ (MetOpen‘𝐷))
253		eqid 2765	. . . . . . . . . . 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))
254	253	fmpox 8070	. . . . . . . . . 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))
255	252, 254	sylib 221	. . . . . . . . 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))
256	255	frnd 6718	. . . . . . . 8 ⊢ (𝜑 → ran (𝑚 ∈ 𝐼, 𝑢 ∈ ((TopOpen ∘ 𝑅)‘𝑚) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐼 ∪ ((TopOpen ∘ 𝑅)‘𝑛) ↦ (𝑤‘𝑚)) “ 𝑢)) ⊆ (MetOpen‘𝐷))
257	179, 256	unssd 4145	. . . . . . 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))
258		fiss 9391	. . . . . . 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)))
259	156, 257, 258	syl2anc 596	. . . . . 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)))
260	153, 259	eqsstrd 3972	. . . . 5 ⊢ (𝜑 → 𝐶 ⊆ (fi‘(MetOpen‘𝐷)))
261		fitop 23109	. . . . . . 7 ⊢ ((MetOpen‘𝐷) ∈ Top → (fi‘(MetOpen‘𝐷)) = (MetOpen‘𝐷))
262	156, 261	syl 18	. . . . . 6 ⊢ (𝜑 → (fi‘(MetOpen‘𝐷)) = (MetOpen‘𝐷))
263	154	mopnval 24648	. . . . . . . 8 ⊢ (𝐷 ∈ (∞Met‘𝐵) → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
264	17, 263	syl 18	. . . . . . 7 ⊢ (𝜑 → (MetOpen‘𝐷) = (topGen‘ran (ball‘𝐷)))
265		tgdif0 23201	. . . . . . 7 ⊢ (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘ran (ball‘𝐷))
266	264, 265	eqtr4di 2818	. . . . . 6 ⊢ (𝜑 → (MetOpen‘𝐷) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
267	262, 266	eqtrd 2800	. . . . 5 ⊢ (𝜑 → (fi‘(MetOpen‘𝐷)) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
268	260, 267	sseqtrd 3974	. . . 4 ⊢ (𝜑 → 𝐶 ⊆ (topGen‘(ran (ball‘𝐷) ∖ {∅})))
269		2basgen 23199	. . . 4 ⊢ (((ran (ball‘𝐷) ∖ {∅}) ⊆ 𝐶 ∧ 𝐶 ⊆ (topGen‘(ran (ball‘𝐷) ∖ {∅}))) → (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘𝐶))
270	136, 268, 269	syl2anc 596	. . 3 ⊢ (𝜑 → (topGen‘(ran (ball‘𝐷) ∖ {∅})) = (topGen‘𝐶))
271	11, 270	eqtr4d 2803	. 2 ⊢ (𝜑 → (∏t‘(TopOpen ∘ 𝑅)) = (topGen‘(ran (ball‘𝐷) ∖ {∅})))
272		prdsxms.j	. . 3 ⊢ 𝐽 = (TopOpen‘𝑌)
273	13, 14, 1, 4, 272	prdstopn 23838	. 2 ⊢ (𝜑 → 𝐽 = (∏t‘(TopOpen ∘ 𝑅)))
274	271, 273, 266	3eqtr4d 2810	1 ⊢ (𝜑 → 𝐽 = (MetOpen‘𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  {crab 3418  Vcvv 3457   ∖ cdif 3903   ∪ cun 3904   ⊆ wss 3906  ∅c0 4286  {csn 4591  ∪ cuni 4874  ∪ ciun 4958   class class class wbr 5111   ↦ cmpt 5194   × cxp 5661  ^◡ccnv 5662  ran crn 5664   ↾ cres 5665   “ cima 5666   ∘ ccom 5667   Fn wfn 6535  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  Xcixp 8901  Fincfn 8949  ficfi 9377  0cc0 11117  ℝ^*cxr 11259   < clt 11260  ℝ⁺crp 13034  Basecbs 17293  distcds 17343  TopOpenctopn 17498  topGenctg 17514  ∏_tcpt 17515  X_scprds 17522  ∞Metcxmet 21559  ballcbl 21561  MetOpencmopn 21564  Topctop 23102  TopOnctopon 23119  TopSpctps 23141  ∞MetSpcxms 24527

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11173  ax-resscn 11174  ax-1cn 11175  ax-icn 11176  ax-addcl 11177  ax-addrcl 11178  ax-mulcl 11179  ax-mulrcl 11180  ax-mulcom 11181  ax-addass 11182  ax-mulass 11183  ax-distr 11184  ax-i2m1 11185  ax-1ne0 11186  ax-1rid 11187  ax-rnegex 11188  ax-rrecex 11189  ax-cnre 11190  ax-pre-lttri 11191  ax-pre-lttrn 11192  ax-pre-ltadd 11193  ax-pre-mulgt0 11194  ax-pre-sup 11195

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4875  df-int 4915  df-iun 4960  df-iin 4961  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-1o 8459  df-2o 8460  df-er 8700  df-map 8832  df-ixp 8902  df-en 8950  df-dom 8951  df-sdom 8952  df-fin 8953  df-fi 9378  df-sup 9409  df-inf 9410  df-pnf 11262  df-mnf 11263  df-xr 11264  df-ltxr 11265  df-le 11266  df-sub 11460  df-neg 11461  df-div 11889  df-nn 12251  df-2 12320  df-3 12321  df-4 12322  df-5 12323  df-6 12324  df-7 12325  df-8 12326  df-9 12327  df-n0 12522  df-z 12609  df-dec 12730  df-uz 12881  df-q 12991  df-rp 13035  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-icc 13397  df-fz 13554  df-struct 17231  df-slot 17266  df-ndx 17278  df-base 17294  df-plusg 17347  df-mulr 17348  df-sca 17350  df-vsca 17351  df-ip 17352  df-tset 17353  df-ple 17354  df-ds 17356  df-hom 17358  df-cco 17359  df-rest 17499  df-topn 17500  df-topgen 17520  df-pt 17521  df-prds 17524  df-psmet 21566  df-xmet 21567  df-bl 21569  df-mopn 21570  df-top 23103  df-topon 23120  df-topsp 23142  df-bases 23155  df-xms 24530

This theorem is used by:  prdsxms  24740