Theorem prdstotbnd 37801

Description: The product metric over finite index set is totally bounded if all the factors are totally bounded. (Contributed by Mario Carneiro, 20-Sep-2015.)

Hypotheses

Ref	Expression
prdsbnd.y	⊢ 𝑌 = (𝑆Xs𝑅)
prdsbnd.b	⊢ 𝐵 = (Base‘𝑌)
prdsbnd.v	⊢ 𝑉 = (Base‘(𝑅‘𝑥))
prdsbnd.e	⊢ 𝐸 = ((dist‘(𝑅‘𝑥)) ↾ (𝑉 × 𝑉))
prdsbnd.d	⊢ 𝐷 = (dist‘𝑌)
prdsbnd.s	⊢ (𝜑 → 𝑆 ∈ 𝑊)
prdsbnd.i	⊢ (𝜑 → 𝐼 ∈ Fin)
prdsbnd.r	⊢ (𝜑 → 𝑅 Fn 𝐼)
prdstotbnd.m	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (TotBnd‘𝑉))

Assertion

Ref	Expression
prdstotbnd	⊢ (𝜑 → 𝐷 ∈ (TotBnd‘𝐵))

Distinct variable groups:   𝑥,𝑅   𝑥,𝐵   𝜑,𝑥   𝑥,𝐼   𝑥,𝑆   𝑥,𝑌

Allowed substitution hints:   𝐷(𝑥)   𝐸(𝑥)   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem prdstotbnd

Dummy variables 𝑧 𝑟 𝑓 𝑔 𝑣 𝑦 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2737	. . . 4 ⊢ (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))) = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
2		eqid 2737	. . . 4 ⊢ (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
3		prdsbnd.v	. . . 4 ⊢ 𝑉 = (Base‘(𝑅‘𝑥))
4		prdsbnd.e	. . . 4 ⊢ 𝐸 = ((dist‘(𝑅‘𝑥)) ↾ (𝑉 × 𝑉))
5		eqid 2737	. . . 4 ⊢ (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) = (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
6		prdsbnd.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
7		prdsbnd.i	. . . 4 ⊢ (𝜑 → 𝐼 ∈ Fin)
8		fvexd 6921	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ V)
9		prdstotbnd.m	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (TotBnd‘𝑉))
10		totbndmet 37779	. . . . 5 ⊢ (𝐸 ∈ (TotBnd‘𝑉) → 𝐸 ∈ (Met‘𝑉))
11	9, 10	syl 17	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (Met‘𝑉))
12	1, 2, 3, 4, 5, 6, 7, 8, 11	prdsmet 24380	. . 3 ⊢ (𝜑 → (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))) ∈ (Met‘(Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))))
13		prdsbnd.d	. . . 4 ⊢ 𝐷 = (dist‘𝑌)
14		prdsbnd.y	. . . . . 6 ⊢ 𝑌 = (𝑆Xs𝑅)
15		prdsbnd.r	. . . . . . . 8 ⊢ (𝜑 → 𝑅 Fn 𝐼)
16		dffn5 6967	. . . . . . . 8 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅 = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
17	15, 16	sylib 218	. . . . . . 7 ⊢ (𝜑 → 𝑅 = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
18	17	oveq2d 7447	. . . . . 6 ⊢ (𝜑 → (𝑆Xs𝑅) = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
19	14, 18	eqtrid 2789	. . . . 5 ⊢ (𝜑 → 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
20	19	fveq2d 6910	. . . 4 ⊢ (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
21	13, 20	eqtrid 2789	. . 3 ⊢ (𝜑 → 𝐷 = (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
22		prdsbnd.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑌)
23	19	fveq2d 6910	. . . . 5 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
24	22, 23	eqtrid 2789	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
25	24	fveq2d 6910	. . 3 ⊢ (𝜑 → (Met‘𝐵) = (Met‘(Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))))
26	12, 21, 25	3eltr4d 2856	. 2 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝐵))
27	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → 𝐼 ∈ Fin)
28		istotbnd3 37778	. . . . . . . . . . 11 ⊢ (𝐸 ∈ (TotBnd‘𝑉) ↔ (𝐸 ∈ (Met‘𝑉) ∧ ∀𝑟 ∈ ℝ+ ∃𝑤 ∈ (𝒫 𝑉 ∩ Fin)∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
29	28	simprbi 496	. . . . . . . . . 10 ⊢ (𝐸 ∈ (TotBnd‘𝑉) → ∀𝑟 ∈ ℝ+ ∃𝑤 ∈ (𝒫 𝑉 ∩ Fin)∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)
30	9, 29	syl 17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → ∀𝑟 ∈ ℝ+ ∃𝑤 ∈ (𝒫 𝑉 ∩ Fin)∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)
31	30	r19.21bi 3251	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑟 ∈ ℝ+) → ∃𝑤 ∈ (𝒫 𝑉 ∩ Fin)∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)
32		df-rex 3071	. . . . . . . . 9 ⊢ (∃𝑤 ∈ (𝒫 𝑉 ∩ Fin)∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉 ↔ ∃𝑤(𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
33		rexv 3509	. . . . . . . . 9 ⊢ (∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉) ↔ ∃𝑤(𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
34	32, 33	bitr4i 278	. . . . . . . 8 ⊢ (∃𝑤 ∈ (𝒫 𝑉 ∩ Fin)∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉 ↔ ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
35	31, 34	sylib 218	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑟 ∈ ℝ+) → ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
36	35	an32s 652	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑥 ∈ 𝐼) → ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
37	36	ralrimiva 3146	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → ∀𝑥 ∈ 𝐼 ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
38		eleq1 2829	. . . . . . 7 ⊢ (𝑤 = (𝑓‘𝑥) → (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ↔ (𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin)))
39		iuneq1 5008	. . . . . . . 8 ⊢ (𝑤 = (𝑓‘𝑥) → ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟))
40	39	eqeq1d 2739	. . . . . . 7 ⊢ (𝑤 = (𝑓‘𝑥) → (∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉 ↔ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))
41	38, 40	anbi12d 632	. . . . . 6 ⊢ (𝑤 = (𝑓‘𝑥) → ((𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉) ↔ ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉)))
42	41	ac6sfi 9320	. . . . 5 ⊢ ((𝐼 ∈ Fin ∧ ∀𝑥 ∈ 𝐼 ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)) → ∃𝑓(𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉)))
43	27, 37, 42	syl2anc 584	. . . 4 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → ∃𝑓(𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉)))
44		elfpw 9394	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ↔ ((𝑓‘𝑥) ⊆ 𝑉 ∧ (𝑓‘𝑥) ∈ Fin))
45	44	simplbi 497	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) → (𝑓‘𝑥) ⊆ 𝑉)
46	45	adantr 480	. . . . . . . . . 10 ⊢ (((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → (𝑓‘𝑥) ⊆ 𝑉)
47	46	ralimi 3083	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ 𝑉)
48	47	ad2antll 729	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ 𝑉)
49		ss2ixp 8950	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ 𝑉 → X𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ X𝑥 ∈ 𝐼 𝑉)
50	48, 49	syl 17	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → X𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ X𝑥 ∈ 𝐼 𝑉)
51		fnfi 9218	. . . . . . . . . . 11 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) → 𝑅 ∈ Fin)
52	15, 7, 51	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Fin)
53	15	fndmd 6673	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑅 = 𝐼)
54	14, 6, 52, 22, 53	prdsbas 17502	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
55	3	rgenw 3065	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ 𝐼 𝑉 = (Base‘(𝑅‘𝑥))
56		ixpeq2 8951	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐼 𝑉 = (Base‘(𝑅‘𝑥)) → X𝑥 ∈ 𝐼 𝑉 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
57	55, 56	ax-mp 5	. . . . . . . . 9 ⊢ X𝑥 ∈ 𝐼 𝑉 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))
58	54, 57	eqtr4di 2795	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 𝑉)
59	58	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → 𝐵 = X𝑥 ∈ 𝐼 𝑉)
60	50, 59	sseqtrrd 4021	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → X𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ 𝐵)
61	27	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → 𝐼 ∈ Fin)
62	44	simprbi 496	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) → (𝑓‘𝑥) ∈ Fin)
63	62	adantr 480	. . . . . . . . 9 ⊢ (((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → (𝑓‘𝑥) ∈ Fin)
64	63	ralimi 3083	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ Fin)
65	64	ad2antll 729	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ Fin)
66		ixpfi 9389	. . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ Fin) → X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ Fin)
67	61, 65, 66	syl2anc 584	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ Fin)
68		elfpw 9394	. . . . . 6 ⊢ (X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ (𝒫 𝐵 ∩ Fin) ↔ (X𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ 𝐵 ∧ X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ Fin))
69	60, 67, 68	sylanbrc 583	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ (𝒫 𝐵 ∩ Fin))
70		metxmet 24344	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (Met‘𝐵) → 𝐷 ∈ (∞Met‘𝐵))
71	26, 70	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))
72		rpxr 13044	. . . . . . . . . 10 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
73		blssm 24428	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
74	73	3expa 1119	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
75	74	an32s 652	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑟 ∈ ℝ*) ∧ 𝑦 ∈ 𝐵) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
76	75	ralrimiva 3146	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑟 ∈ ℝ*) → ∀𝑦 ∈ 𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
77	71, 72, 76	syl2an 596	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → ∀𝑦 ∈ 𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
78	77	adantr 480	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∀𝑦 ∈ 𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
79		ssralv 4052	. . . . . . . 8 ⊢ (X𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵 → ∀𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵))
80	60, 78, 79	sylc 65	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∀𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
81		iunss 5045	. . . . . . 7 ⊢ (∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵 ↔ ∀𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
82	80, 81	sylibr 234	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
83	61	adantr 480	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → 𝐼 ∈ Fin)
84	59	eleq2d 2827	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (𝑔 ∈ 𝐵 ↔ 𝑔 ∈ X𝑥 ∈ 𝐼 𝑉))
85		vex 3484	. . . . . . . . . . . . . . . 16 ⊢ 𝑔 ∈ V
86	85	elixp 8944	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐼 𝑉 ↔ (𝑔 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉))
87	86	simprbi 496	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐼 𝑉 → ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉)
88		df-rex 3071	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑧 ∈ (𝑓‘𝑥)(𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧(𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))
89		eliun 4995	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔‘𝑥) ∈ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧 ∈ (𝑓‘𝑥)(𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟))
90		rexv 3509	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) ↔ ∃𝑧(𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))
91	88, 89, 90	3bitr4i 303	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔‘𝑥) ∈ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))
92		eleq2 2830	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → ((𝑔‘𝑥) ∈ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) ↔ (𝑔‘𝑥) ∈ 𝑉))
93	91, 92	bitr3id 285	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → (∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) ↔ (𝑔‘𝑥) ∈ 𝑉))
94	93	biimprd 248	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → ((𝑔‘𝑥) ∈ 𝑉 → ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟))))
95	94	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → ((𝑔‘𝑥) ∈ 𝑉 → ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟))))
96	95	ral2imi 3085	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → (∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉 → ∀𝑥 ∈ 𝐼 ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟))))
97	96	ad2antll 729	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉 → ∀𝑥 ∈ 𝐼 ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟))))
98	87, 97	syl5 34	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (𝑔 ∈ X𝑥 ∈ 𝐼 𝑉 → ∀𝑥 ∈ 𝐼 ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟))))
99	84, 98	sylbid 240	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (𝑔 ∈ 𝐵 → ∀𝑥 ∈ 𝐼 ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟))))
100	99	imp 406	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → ∀𝑥 ∈ 𝐼 ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))
101		eleq1 2829	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑦‘𝑥) → (𝑧 ∈ (𝑓‘𝑥) ↔ (𝑦‘𝑥) ∈ (𝑓‘𝑥)))
102		oveq1 7438	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑦‘𝑥) → (𝑧(ball‘𝐸)𝑟) = ((𝑦‘𝑥)(ball‘𝐸)𝑟))
103	102	eleq2d 2827	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑦‘𝑥) → ((𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟) ↔ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))
104	101, 103	anbi12d 632	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑦‘𝑥) → ((𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) ↔ ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))))
105	104	ac6sfi 9320	. . . . . . . . . . 11 ⊢ ((𝐼 ∈ Fin ∧ ∀𝑥 ∈ 𝐼 ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟))) → ∃𝑦(𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))))
106	83, 100, 105	syl2anc 584	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → ∃𝑦(𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))))
107		ffn 6736	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦:𝐼⟶V → 𝑦 Fn 𝐼)
108		simpl 482	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)) → (𝑦‘𝑥) ∈ (𝑓‘𝑥))
109	108	ralimi 3083	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)) → ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (𝑓‘𝑥))
110	107, 109	anim12i 613	. . . . . . . . . . . . . . . 16 ⊢ ((𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))) → (𝑦 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (𝑓‘𝑥)))
111		vex 3484	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
112	111	elixp 8944	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 (𝑓‘𝑥) ↔ (𝑦 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (𝑓‘𝑥)))
113	110, 112	sylibr 234	. . . . . . . . . . . . . . 15 ⊢ ((𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))) → 𝑦 ∈ X𝑥 ∈ 𝐼 (𝑓‘𝑥))
114	113	adantl 481	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝑦 ∈ X𝑥 ∈ 𝐼 (𝑓‘𝑥))
115	84	biimpa 476	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ X𝑥 ∈ 𝐼 𝑉)
116		ixpfn 8943	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐼 𝑉 → 𝑔 Fn 𝐼)
117	115, 116	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → 𝑔 Fn 𝐼)
118	117	adantr 480	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝑔 Fn 𝐼)
119		simpr 484	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)) → (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))
120	119	ralimi 3083	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)) → ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))
121	120	ad2antll 729	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))
122	85	elixp 8944	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐼 ((𝑦‘𝑥)(ball‘𝐸)𝑟) ↔ (𝑔 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))
123	118, 121, 122	sylanbrc 583	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝑔 ∈ X𝑥 ∈ 𝐼 ((𝑦‘𝑥)(ball‘𝐸)𝑟))
124		simp-4l 783	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝜑)
125	50	ad2antrr 726	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → X𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ X𝑥 ∈ 𝐼 𝑉)
126	125, 114	sseldd 3984	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝑦 ∈ X𝑥 ∈ 𝐼 𝑉)
127	124, 58	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝐵 = X𝑥 ∈ 𝐼 𝑉)
128	126, 127	eleqtrrd 2844	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝑦 ∈ 𝐵)
129		simp-4r 784	. . . . . . . . . . . . . . . 16 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝑟 ∈ ℝ+)
130		fveq2 6906	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑥 → (𝑅‘𝑦) = (𝑅‘𝑥))
131	130	cbvmptv 5255	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)) = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))
132	131	oveq2i 7442	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))) = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
133	19, 132	eqtr4di 2795	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑌 = (𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))
134	133	fveq2d 6910	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
135	13, 134	eqtrid 2789	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐷 = (dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
136	135	fveq2d 6910	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ball‘𝐷) = (ball‘(dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))))
137	136	oveqdr 7459	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘(dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))𝑟))
138		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))) = (Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))
139		eqid 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))) = (dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))
140	6	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝑆 ∈ 𝑊)
141	7	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝐼 ∈ Fin)
142		fvexd 6921	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ V)
143		metxmet 24344	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸 ∈ (Met‘𝑉) → 𝐸 ∈ (∞Met‘𝑉))
144	11, 143	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
145	144	adantlr 715	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
146		simprl 771	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝑦 ∈ 𝐵)
147	133	fveq2d 6910	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
148	22, 147	eqtrid 2789	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
149	148	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝐵 = (Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
150	146, 149	eleqtrd 2843	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝑦 ∈ (Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
151	72	ad2antll 729	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ*)
152		rpgt0 13047	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ ℝ+ → 0 < 𝑟)
153	152	ad2antll 729	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 0 < 𝑟)
154	132, 138, 3, 4, 139, 140, 141, 142, 145, 150, 151, 153	prdsbl 24504	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → (𝑦(ball‘(dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))𝑟) = X𝑥 ∈ 𝐼 ((𝑦‘𝑥)(ball‘𝐸)𝑟))
155	137, 154	eqtrd 2777	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → (𝑦(ball‘𝐷)𝑟) = X𝑥 ∈ 𝐼 ((𝑦‘𝑥)(ball‘𝐸)𝑟))
156	124, 128, 129, 155	syl12anc 837	. . . . . . . . . . . . . . 15 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → (𝑦(ball‘𝐷)𝑟) = X𝑥 ∈ 𝐼 ((𝑦‘𝑥)(ball‘𝐸)𝑟))
157	123, 156	eleqtrrd 2844	. . . . . . . . . . . . . 14 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → 𝑔 ∈ (𝑦(ball‘𝐷)𝑟))
158	114, 157	jca 511	. . . . . . . . . . . . 13 ⊢ (((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) ∧ (𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))) → (𝑦 ∈ X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∧ 𝑔 ∈ (𝑦(ball‘𝐷)𝑟)))
159	158	ex 412	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → ((𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))) → (𝑦 ∈ X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∧ 𝑔 ∈ (𝑦(ball‘𝐷)𝑟))))
160	159	eximdv 1917	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → (∃𝑦(𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))) → ∃𝑦(𝑦 ∈ X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∧ 𝑔 ∈ (𝑦(ball‘𝐷)𝑟))))
161		df-rex 3071	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦(𝑦 ∈ X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∧ 𝑔 ∈ (𝑦(ball‘𝐷)𝑟)))
162	160, 161	imbitrrdi 252	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → (∃𝑦(𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))) → ∃𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟)))
163	106, 162	mpd 15	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) ∧ 𝑔 ∈ 𝐵) → ∃𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟))
164	163	ex 412	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (𝑔 ∈ 𝐵 → ∃𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟)))
165		eliun 4995	. . . . . . . 8 ⊢ (𝑔 ∈ ∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟))
166	164, 165	imbitrrdi 252	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → (𝑔 ∈ 𝐵 → 𝑔 ∈ ∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟)))
167	166	ssrdv 3989	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → 𝐵 ⊆ ∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟))
168	82, 167	eqssd 4001	. . . . 5 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) = 𝐵)
169		iuneq1 5008	. . . . . . 7 ⊢ (𝑣 = X𝑥 ∈ 𝐼 (𝑓‘𝑥) → ∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = ∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟))
170	169	eqeq1d 2739	. . . . . 6 ⊢ (𝑣 = X𝑥 ∈ 𝐼 (𝑓‘𝑥) → (∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵 ↔ ∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) = 𝐵))
171	170	rspcev 3622	. . . . 5 ⊢ ((X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ (𝒫 𝐵 ∩ Fin) ∧ ∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) = 𝐵) → ∃𝑣 ∈ (𝒫 𝐵 ∩ Fin)∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵)
172	69, 168, 171	syl2anc 584	. . . 4 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → ∃𝑣 ∈ (𝒫 𝐵 ∩ Fin)∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵)
173	43, 172	exlimddv 1935	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → ∃𝑣 ∈ (𝒫 𝐵 ∩ Fin)∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵)
174	173	ralrimiva 3146	. 2 ⊢ (𝜑 → ∀𝑟 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝐵 ∩ Fin)∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵)
175		istotbnd3 37778	. 2 ⊢ (𝐷 ∈ (TotBnd‘𝐵) ↔ (𝐷 ∈ (Met‘𝐵) ∧ ∀𝑟 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝐵 ∩ Fin)∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵))
176	26, 174, 175	sylanbrc 583	1 ⊢ (𝜑 → 𝐷 ∈ (TotBnd‘𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ∩ cin 3950   ⊆ wss 3951  𝒫 cpw 4600  ∪ ciun 4991   class class class wbr 5143   ↦ cmpt 5225   × cxp 5683   ↾ cres 5687   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  Xcixp 8937  Fincfn 8985  0cc0 11155  ℝ^*cxr 11294   < clt 11295  ℝ⁺crp 13034  Basecbs 17247  distcds 17306  X_scprds 17490  ∞Metcxmet 21349  Metcmet 21350  ballcbl 21351  TotBndctotbnd 37773

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232  ax-pre-sup 11233

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-pm 8869  df-ixp 8938  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-sup 9482  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-rp 13035  df-xneg 13154  df-xadd 13155  df-xmul 13156  df-icc 13394  df-fz 13548  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-plusg 17310  df-mulr 17311  df-sca 17313  df-vsca 17314  df-ip 17315  df-tset 17316  df-ple 17317  df-ds 17319  df-hom 17321  df-cco 17322  df-prds 17492  df-psmet 21356  df-xmet 21357  df-met 21358  df-bl 21359  df-totbnd 37775

This theorem is referenced by:  prdsbnd2  37802