Theorem prdstotbnd 35074

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 2823	. . . 4 ⊢ (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))) = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
2		eqid 2823	. . . 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 2823	. . . 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 6687	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ V)
9		prdstotbnd.m	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (TotBnd‘𝑉))
10		totbndmet 35052	. . . . 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 22982	. . 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 6726	. . . . . . . 8 ⊢ (𝑅 Fn 𝐼 ↔ 𝑅 = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
17	15, 16	sylib 220	. . . . . . 7 ⊢ (𝜑 → 𝑅 = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
18	17	oveq2d 7174	. . . . . 6 ⊢ (𝜑 → (𝑆Xs𝑅) = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
19	14, 18	syl5eq 2870	. . . . 5 ⊢ (𝜑 → 𝑌 = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))
20	19	fveq2d 6676	. . . 4 ⊢ (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
21	13, 20	syl5eq 2870	. . 3 ⊢ (𝜑 → 𝐷 = (dist‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
22		prdsbnd.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑌)
23	19	fveq2d 6676	. . . . 5 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
24	22, 23	syl5eq 2870	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))))
25	24	fveq2d 6676	. . 3 ⊢ (𝜑 → (Met‘𝐵) = (Met‘(Base‘(𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))))))
26	12, 21, 25	3eltr4d 2930	. 2 ⊢ (𝜑 → 𝐷 ∈ (Met‘𝐵))
27	7	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → 𝐼 ∈ Fin)
28		istotbnd3 35051	. . . . . . . . . . 11 ⊢ (𝐸 ∈ (TotBnd‘𝑉) ↔ (𝐸 ∈ (Met‘𝑉) ∧ ∀𝑟 ∈ ℝ+ ∃𝑤 ∈ (𝒫 𝑉 ∩ Fin)∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
29	28	simprbi 499	. . . . . . . . . 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 3210	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑟 ∈ ℝ+) → ∃𝑤 ∈ (𝒫 𝑉 ∩ Fin)∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉)
32		df-rex 3146	. . . . . . . . 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 3522	. . . . . . . . 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 280	. . . . . . . 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 220	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐼) ∧ 𝑟 ∈ ℝ+) → ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
36	35	an32s 650	. . . . . 6 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ 𝑥 ∈ 𝐼) → ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
37	36	ralrimiva 3184	. . . . 5 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → ∀𝑥 ∈ 𝐼 ∃𝑤 ∈ V (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = 𝑉))
38		eleq1 2902	. . . . . . 7 ⊢ (𝑤 = (𝑓‘𝑥) → (𝑤 ∈ (𝒫 𝑉 ∩ Fin) ↔ (𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin)))
39		iuneq1 4937	. . . . . . . 8 ⊢ (𝑤 = (𝑓‘𝑥) → ∪ 𝑧 ∈ 𝑤 (𝑧(ball‘𝐸)𝑟) = ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟))
40	39	eqeq1d 2825	. . . . . . 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 8764	. . . . 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 586	. . . 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 8828	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ↔ ((𝑓‘𝑥) ⊆ 𝑉 ∧ (𝑓‘𝑥) ∈ Fin))
45	44	simplbi 500	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) → (𝑓‘𝑥) ⊆ 𝑉)
46	45	adantr 483	. . . . . . . . . 10 ⊢ (((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → (𝑓‘𝑥) ⊆ 𝑉)
47	46	ralimi 3162	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ 𝑉)
48	47	ad2antll 727	. . . . . . . 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 8476	. . . . . . . 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 8798	. . . . . . . . . . 11 ⊢ ((𝑅 Fn 𝐼 ∧ 𝐼 ∈ Fin) → 𝑅 ∈ Fin)
52	15, 7, 51	syl2anc 586	. . . . . . . . . 10 ⊢ (𝜑 → 𝑅 ∈ Fin)
53		fndm 6457	. . . . . . . . . . 11 ⊢ (𝑅 Fn 𝐼 → dom 𝑅 = 𝐼)
54	15, 53	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝑅 = 𝐼)
55	14, 6, 52, 22, 54	prdsbas 16732	. . . . . . . . 9 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
56	3	rgenw 3152	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ 𝐼 𝑉 = (Base‘(𝑅‘𝑥))
57		ixpeq2 8477	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝐼 𝑉 = (Base‘(𝑅‘𝑥)) → X𝑥 ∈ 𝐼 𝑉 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥)))
58	56, 57	ax-mp 5	. . . . . . . . 9 ⊢ X𝑥 ∈ 𝐼 𝑉 = X𝑥 ∈ 𝐼 (Base‘(𝑅‘𝑥))
59	55, 58	syl6eqr 2876	. . . . . . . 8 ⊢ (𝜑 → 𝐵 = X𝑥 ∈ 𝐼 𝑉)
60	59	ad2antrr 724	. . . . . . 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)
61	50, 60	sseqtrrd 4010	. . . . . 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)
62	27	adantr 483	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑟 ∈ ℝ+) ∧ (𝑓:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉))) → 𝐼 ∈ Fin)
63	44	simprbi 499	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) → (𝑓‘𝑥) ∈ Fin)
64	63	adantr 483	. . . . . . . . 9 ⊢ (((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → (𝑓‘𝑥) ∈ Fin)
65	64	ralimi 3162	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐼 ((𝑓‘𝑥) ∈ (𝒫 𝑉 ∩ Fin) ∧ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉) → ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ Fin)
66	65	ad2antll 727	. . . . . . 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)
67		ixpfi 8823	. . . . . . 7 ⊢ ((𝐼 ∈ Fin ∧ ∀𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ Fin) → X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ Fin)
68	62, 66, 67	syl2anc 586	. . . . . 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)
69		elfpw 8828	. . . . . 6 ⊢ (X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ (𝒫 𝐵 ∩ Fin) ↔ (X𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ 𝐵 ∧ X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∈ Fin))
70	61, 68, 69	sylanbrc 585	. . . . 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))
71		metxmet 22946	. . . . . . . . . . 11 ⊢ (𝐷 ∈ (Met‘𝐵) → 𝐷 ∈ (∞Met‘𝐵))
72	26, 71	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝐵))
73		rpxr 12401	. . . . . . . . . 10 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
74		blssm 23030	. . . . . . . . . . . . 13 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
75	74	3expa 1114	. . . . . . . . . . . 12 ⊢ (((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑦 ∈ 𝐵) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
76	75	an32s 650	. . . . . . . . . . 11 ⊢ (((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑟 ∈ ℝ*) ∧ 𝑦 ∈ 𝐵) → (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
77	76	ralrimiva 3184	. . . . . . . . . 10 ⊢ ((𝐷 ∈ (∞Met‘𝐵) ∧ 𝑟 ∈ ℝ*) → ∀𝑦 ∈ 𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
78	72, 73, 77	syl2an 597	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → ∀𝑦 ∈ 𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
79	78	adantr 483	. . . . . . . 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)
80		ssralv 4035	. . . . . . . 8 ⊢ (X𝑥 ∈ 𝐼 (𝑓‘𝑥) ⊆ 𝐵 → (∀𝑦 ∈ 𝐵 (𝑦(ball‘𝐷)𝑟) ⊆ 𝐵 → ∀𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵))
81	61, 79, 80	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)
82		iunss 4971	. . . . . . 7 ⊢ (∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵 ↔ ∀𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) ⊆ 𝐵)
83	81, 82	sylibr 236	. . . . . 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)
84	62	adantr 483	. . . . . . . . . . 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)
85	60	eleq2d 2900	. . . . . . . . . . . . 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))
86		vex 3499	. . . . . . . . . . . . . . . 16 ⊢ 𝑔 ∈ V
87	86	elixp 8470	. . . . . . . . . . . . . . 15 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐼 𝑉 ↔ (𝑔 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉))
88	87	simprbi 499	. . . . . . . . . . . . . 14 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐼 𝑉 → ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ 𝑉)
89		df-rex 3146	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑧 ∈ (𝑓‘𝑥)(𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧(𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))
90		eliun 4925	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑔‘𝑥) ∈ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧 ∈ (𝑓‘𝑥)(𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟))
91		rexv 3522	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) ↔ ∃𝑧(𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))
92	89, 90, 91	3bitr4i 305	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑔‘𝑥) ∈ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) ↔ ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)))
93		eleq2 2903	. . . . . . . . . . . . . . . . . . 19 ⊢ (∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → ((𝑔‘𝑥) ∈ ∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) ↔ (𝑔‘𝑥) ∈ 𝑉))
94	92, 93	syl5bbr 287	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → (∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟)) ↔ (𝑔‘𝑥) ∈ 𝑉))
95	94	biimprd 250	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑧 ∈ (𝑓‘𝑥)(𝑧(ball‘𝐸)𝑟) = 𝑉 → ((𝑔‘𝑥) ∈ 𝑉 → ∃𝑧 ∈ V (𝑧 ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟))))
96	95	adantl 484	. . . . . . . . . . . . . . . 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))))
97	96	ral2imi 3158	. . . . . . . . . . . . . . 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))))
98	97	ad2antll 727	. . . . . . . . . . . . . 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))))
99	88, 98	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))))
100	85, 99	sylbid 242	. . . . . . . . . . . 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))))
101	100	imp 409	. . . . . . . . . . 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)))
102		eleq1 2902	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑦‘𝑥) → (𝑧 ∈ (𝑓‘𝑥) ↔ (𝑦‘𝑥) ∈ (𝑓‘𝑥)))
103		oveq1 7165	. . . . . . . . . . . . . 14 ⊢ (𝑧 = (𝑦‘𝑥) → (𝑧(ball‘𝐸)𝑟) = ((𝑦‘𝑥)(ball‘𝐸)𝑟))
104	103	eleq2d 2900	. . . . . . . . . . . . 13 ⊢ (𝑧 = (𝑦‘𝑥) → ((𝑔‘𝑥) ∈ (𝑧(ball‘𝐸)𝑟) ↔ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))
105	102, 104	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))))
106	105	ac6sfi 8764	. . . . . . . . . . 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))))
107	84, 101, 106	syl2anc 586	. . . . . . . . . 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))))
108		ffn 6516	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦:𝐼⟶V → 𝑦 Fn 𝐼)
109		simpl 485	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)) → (𝑦‘𝑥) ∈ (𝑓‘𝑥))
110	109	ralimi 3162	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)) → ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (𝑓‘𝑥))
111	108, 110	anim12i 614	. . . . . . . . . . . . . . . 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)))
112		vex 3499	. . . . . . . . . . . . . . . . 17 ⊢ 𝑦 ∈ V
113	112	elixp 8470	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ X𝑥 ∈ 𝐼 (𝑓‘𝑥) ↔ (𝑦 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑦‘𝑥) ∈ (𝑓‘𝑥)))
114	111, 113	sylibr 236	. . . . . . . . . . . . . . 15 ⊢ ((𝑦:𝐼⟶V ∧ ∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))) → 𝑦 ∈ X𝑥 ∈ 𝐼 (𝑓‘𝑥))
115	114	adantl 484	. . . . . . . . . . . . . 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))
116	85	biimpa 479	. . . . . . . . . . . . . . . . . 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)
117		ixpfn 8469	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐼 𝑉 → 𝑔 Fn 𝐼)
118	116, 117	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)
119	118	adantr 483	. . . . . . . . . . . . . . . 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)
120		simpr 487	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)) → (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))
121	120	ralimi 3162	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑥 ∈ 𝐼 ((𝑦‘𝑥) ∈ (𝑓‘𝑥) ∧ (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)) → ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟))
122	121	ad2antll 727	. . . . . . . . . . . . . . . 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))
123	86	elixp 8470	. . . . . . . . . . . . . . . 16 ⊢ (𝑔 ∈ X𝑥 ∈ 𝐼 ((𝑦‘𝑥)(ball‘𝐸)𝑟) ↔ (𝑔 Fn 𝐼 ∧ ∀𝑥 ∈ 𝐼 (𝑔‘𝑥) ∈ ((𝑦‘𝑥)(ball‘𝐸)𝑟)))
124	119, 122, 123	sylanbrc 585	. . . . . . . . . . . . . . 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))
125		simp-4l 781	. . . . . . . . . . . . . . . 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)
126	50	ad2antrr 724	. . . . . . . . . . . . . . . . . 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)
127	126, 115	sseldd 3970	. . . . . . . . . . . . . . . . 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)
128	125, 59	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)
129	127, 128	eleqtrrd 2918	. . . . . . . . . . . . . . . 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)
130		simp-4r 782	. . . . . . . . . . . . . . . 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+)
131		fveq2 6672	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑦 = 𝑥 → (𝑅‘𝑦) = (𝑅‘𝑥))
132	131	cbvmptv 5171	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)) = (𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥))
133	132	oveq2i 7169	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))) = (𝑆Xs(𝑥 ∈ 𝐼 ↦ (𝑅‘𝑥)))
134	19, 133	syl6eqr 2876	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → 𝑌 = (𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))
135	134	fveq2d 6676	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (dist‘𝑌) = (dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
136	13, 135	syl5eq 2870	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐷 = (dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
137	136	fveq2d 6676	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (ball‘𝐷) = (ball‘(dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))))
138	137	oveqdr 7186	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → (𝑦(ball‘𝐷)𝑟) = (𝑦(ball‘(dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))𝑟))
139		eqid 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))) = (Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))
140		eqid 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))) = (dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦))))
141	6	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝑆 ∈ 𝑊)
142	7	adantr 483	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝐼 ∈ Fin)
143		fvexd 6687	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) ∧ 𝑥 ∈ 𝐼) → (𝑅‘𝑥) ∈ V)
144		metxmet 22946	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐸 ∈ (Met‘𝑉) → 𝐸 ∈ (∞Met‘𝑉))
145	11, 144	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
146	145	adantlr 713	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) ∧ 𝑥 ∈ 𝐼) → 𝐸 ∈ (∞Met‘𝑉))
147		simprl 769	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝑦 ∈ 𝐵)
148	134	fveq2d 6676	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝜑 → (Base‘𝑌) = (Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
149	22, 148	syl5eq 2870	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐵 = (Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
150	149	adantr 483	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝐵 = (Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
151	147, 150	eleqtrd 2917	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝑦 ∈ (Base‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))
152	73	ad2antll 727	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 𝑟 ∈ ℝ*)
153		rpgt0 12404	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑟 ∈ ℝ+ → 0 < 𝑟)
154	153	ad2antll 727	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → 0 < 𝑟)
155	133, 139, 3, 4, 140, 141, 142, 143, 146, 151, 152, 154	prdsbl 23103	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → (𝑦(ball‘(dist‘(𝑆Xs(𝑦 ∈ 𝐼 ↦ (𝑅‘𝑦)))))𝑟) = X𝑥 ∈ 𝐼 ((𝑦‘𝑥)(ball‘𝐸)𝑟))
156	138, 155	eqtrd 2858	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝑟 ∈ ℝ+)) → (𝑦(ball‘𝐷)𝑟) = X𝑥 ∈ 𝐼 ((𝑦‘𝑥)(ball‘𝐸)𝑟))
157	125, 129, 130, 156	syl12anc 834	. . . . . . . . . . . . . . 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))
158	124, 157	eleqtrrd 2918	. . . . . . . . . . . . . 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))
159	115, 158	jca 514	. . . . . . . . . . . . 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)))
160	159	ex 415	. . . . . . . . . . . 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))))
161	160	eximdv 1918	. . . . . . . . . . 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))))
162		df-rex 3146	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦(𝑦 ∈ X𝑥 ∈ 𝐼 (𝑓‘𝑥) ∧ 𝑔 ∈ (𝑦(ball‘𝐷)𝑟)))
163	161, 162	syl6ibr 254	. . . . . . . . . 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)))
164	107, 163	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))
165	164	ex 415	. . . . . . . 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)))
166		eliun 4925	. . . . . . . 8 ⊢ (𝑔 ∈ ∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) ↔ ∃𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)𝑔 ∈ (𝑦(ball‘𝐷)𝑟))
167	165, 166	syl6ibr 254	. . . . . . 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)))
168	167	ssrdv 3975	. . . . . 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))
169	83, 168	eqssd 3986	. . . . 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)
170		iuneq1 4937	. . . . . . 7 ⊢ (𝑣 = X𝑥 ∈ 𝐼 (𝑓‘𝑥) → ∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = ∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟))
171	170	eqeq1d 2825	. . . . . 6 ⊢ (𝑣 = X𝑥 ∈ 𝐼 (𝑓‘𝑥) → (∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵 ↔ ∪ 𝑦 ∈ X 𝑥 ∈ 𝐼 (𝑓‘𝑥)(𝑦(ball‘𝐷)𝑟) = 𝐵))
172	171	rspcev 3625	. . . . 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)
173	70, 169, 172	syl2anc 586	. . . 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)
174	43, 173	exlimddv 1936	. . 3 ⊢ ((𝜑 ∧ 𝑟 ∈ ℝ+) → ∃𝑣 ∈ (𝒫 𝐵 ∩ Fin)∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵)
175	174	ralrimiva 3184	. 2 ⊢ (𝜑 → ∀𝑟 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝐵 ∩ Fin)∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵)
176		istotbnd3 35051	. 2 ⊢ (𝐷 ∈ (TotBnd‘𝐵) ↔ (𝐷 ∈ (Met‘𝐵) ∧ ∀𝑟 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝐵 ∩ Fin)∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝐷)𝑟) = 𝐵))
177	26, 175, 176	sylanbrc 585	1 ⊢ (𝜑 → 𝐷 ∈ (TotBnd‘𝐵))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141  Vcvv 3496   ∩ cin 3937   ⊆ wss 3938  𝒫 cpw 4541  ∪ ciun 4921   class class class wbr 5068   ↦ cmpt 5148   × cxp 5555  dom cdm 5557   ↾ cres 5559   Fn wfn 6352  ⟶wf 6353  ‘cfv 6357  (class class class)co 7158  Xcixp 8463  Fincfn 8511  0cc0 10539  ℝ^*cxr 10676   < clt 10677  ℝ⁺crp 12392  Basecbs 16485  distcds 16576  X_scprds 16721  ∞Metcxmet 20532  Metcmet 20533  ballcbl 20534  TotBndctotbnd 35046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-cnex 10595  ax-resscn 10596  ax-1cn 10597  ax-icn 10598  ax-addcl 10599  ax-addrcl 10600  ax-mulcl 10601  ax-mulrcl 10602  ax-mulcom 10603  ax-addass 10604  ax-mulass 10605  ax-distr 10606  ax-i2m1 10607  ax-1ne0 10608  ax-1rid 10609  ax-rnegex 10610  ax-rrecex 10611  ax-cnre 10612  ax-pre-lttri 10613  ax-pre-lttrn 10614  ax-pre-ltadd 10615  ax-pre-mulgt0 10616  ax-pre-sup 10617

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-nel 3126  df-ral 3145  df-rex 3146  df-reu 3147  df-rmo 3148  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-int 4879  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-pred 6150  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-om 7583  df-1st 7691  df-2nd 7692  df-wrecs 7949  df-recs 8010  df-rdg 8048  df-1o 8104  df-2o 8105  df-oadd 8108  df-er 8291  df-map 8410  df-pm 8411  df-ixp 8464  df-en 8512  df-dom 8513  df-sdom 8514  df-fin 8515  df-sup 8908  df-pnf 10679  df-mnf 10680  df-xr 10681  df-ltxr 10682  df-le 10683  df-sub 10874  df-neg 10875  df-div 11300  df-nn 11641  df-2 11703  df-3 11704  df-4 11705  df-5 11706  df-6 11707  df-7 11708  df-8 11709  df-9 11710  df-n0 11901  df-z 11985  df-dec 12102  df-uz 12247  df-rp 12393  df-xneg 12510  df-xadd 12511  df-xmul 12512  df-icc 12748  df-fz 12896  df-struct 16487  df-ndx 16488  df-slot 16489  df-base 16491  df-plusg 16580  df-mulr 16581  df-sca 16583  df-vsca 16584  df-ip 16585  df-tset 16586  df-ple 16587  df-ds 16589  df-hom 16591  df-cco 16592  df-prds 16723  df-psmet 20539  df-xmet 20540  df-met 20541  df-bl 20542  df-totbnd 35048

This theorem is referenced by:  prdsbnd2  35075