Theorem totbndbnd 37783

Description: A totally bounded metric space is bounded. This theorem fails for extended metrics - a bounded extended metric is a metric, but there are totally bounded extended metrics that are not metrics (if we were to weaken istotbnd 37763 to only require that 𝑀 be an extended metric). A counterexample is the discrete extended metric (assigning distinct points distance +∞) on a finite set. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 12-Sep-2015.)

Assertion

Ref	Expression
totbndbnd	⊢ (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Bnd‘𝑋))

Proof of Theorem totbndbnd

Dummy variables 𝑣 𝑑 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		totbndmet 37766	. 2 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Met‘𝑋))
2		1rp 12955	. . 3 ⊢ 1 ∈ ℝ+
3		istotbnd3 37765	. . . 4 ⊢ (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
4	3	simprbi 496	. . 3 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)
5		oveq2 7395	. . . . . . 7 ⊢ (𝑑 = 1 → (𝑥(ball‘𝑀)𝑑) = (𝑥(ball‘𝑀)1))
6	5	iuneq2d 4986	. . . . . 6 ⊢ (𝑑 = 1 → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1))
7	6	eqeq1d 2731	. . . . 5 ⊢ (𝑑 = 1 → (∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 ↔ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
8	7	rexbidv 3157	. . . 4 ⊢ (𝑑 = 1 → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 ↔ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
9	8	rspcv 3584	. . 3 ⊢ (1 ∈ ℝ+ → (∀𝑑 ∈ ℝ+ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋))
10	2, 4, 9	mpsyl 68	. 2 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)
11		simplll 774	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 𝑀 ∈ (Met‘𝑋))
12		elfpw 9305	. . . . . . . . . . . . . 14 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 ⊆ 𝑋 ∧ 𝑣 ∈ Fin))
13	12	simplbi 497	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ⊆ 𝑋)
14	13	ad2antrl 728	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑣 ⊆ 𝑋)
15	14	sselda 3946	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝑋)
16		simpllr 775	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 𝑦 ∈ 𝑋)
17		metcl 24220	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → (𝑧𝑀𝑦) ∈ ℝ)
18	11, 15, 16, 17	syl3anc 1373	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → (𝑧𝑀𝑦) ∈ ℝ)
19		metge0 24233	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) → 0 ≤ (𝑧𝑀𝑦))
20	11, 15, 16, 19	syl3anc 1373	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 0 ≤ (𝑧𝑀𝑦))
21	18, 20	ge0p1rpd 13025	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → ((𝑧𝑀𝑦) + 1) ∈ ℝ+)
22	21	fmpttd 7087	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)):𝑣⟶ℝ+)
23	22	frnd 6696	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ+)
24	12	simprbi 496	. . . . . . . . . 10 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ∈ Fin)
25		mptfi 9302	. . . . . . . . . 10 ⊢ (𝑣 ∈ Fin → (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
26		rnfi 9291	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
27	24, 25, 26	3syl 18	. . . . . . . . 9 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
28	27	ad2antrl 728	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
29		simplr 768	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑦 ∈ 𝑋)
30		simprr 772	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)
31	29, 30	eleqtrrd 2831	. . . . . . . . 9 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑦 ∈ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1))
32		ne0i 4304	. . . . . . . . 9 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) ≠ ∅)
33		dm0rn0 5888	. . . . . . . . . . 11 ⊢ (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅)
34		ovex 7420	. . . . . . . . . . . . . . 15 ⊢ ((𝑧𝑀𝑦) + 1) ∈ V
35		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
36	34, 35	dmmpti 6662	. . . . . . . . . . . . . 14 ⊢ dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = 𝑣
37	36	eqeq1i 2734	. . . . . . . . . . . . 13 ⊢ (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ ↔ 𝑣 = ∅)
38		iuneq1 4972	. . . . . . . . . . . . 13 ⊢ (𝑣 = ∅ → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1))
39	37, 38	sylbi 217	. . . . . . . . . . . 12 ⊢ (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1))
40		0iun 5027	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)1) = ∅
41	39, 40	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = ∅)
42	33, 41	sylbir 235	. . . . . . . . . 10 ⊢ (ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = ∅ → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = ∅)
43	42	necon3i 2957	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) ≠ ∅ → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅)
44	31, 32, 43	3syl 18	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅)
45		rpssre 12959	. . . . . . . . 9 ⊢ ℝ+ ⊆ ℝ
46	23, 45	sstrdi 3959	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ)
47		ltso 11254	. . . . . . . . 9 ⊢ < Or ℝ
48		fisupcl 9421	. . . . . . . . 9 ⊢ (( < Or ℝ ∧ (ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅ ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ)) → sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
49	47, 48	mpan 690	. . . . . . . 8 ⊢ ((ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅ ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ) → sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
50	28, 44, 46, 49	syl3anc 1373	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
51	23, 50	sseldd 3947	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ+)
52		metxmet 24222	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
53	52	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑀 ∈ (∞Met‘𝑋))
54	53	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 𝑀 ∈ (∞Met‘𝑋))
55		1red 11175	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → 1 ∈ ℝ)
56	46, 50	sseldd 3947	. . . . . . . . . . . . 13 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ)
57	56	adantr 480	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ)
58	46	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ)
59	44	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅)
60	28	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
61		fimaxre2 12128	. . . . . . . . . . . . . . 15 ⊢ ((ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin) → ∃𝑑 ∈ ℝ ∀𝑤 ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑤 ≤ 𝑑)
62	58, 60, 61	syl2anc 584	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → ∃𝑑 ∈ ℝ ∀𝑤 ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑤 ≤ 𝑑)
63	35	elrnmpt1 5924	. . . . . . . . . . . . . . . 16 ⊢ ((𝑧 ∈ 𝑣 ∧ ((𝑧𝑀𝑦) + 1) ∈ V) → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
64	34, 63	mpan2 691	. . . . . . . . . . . . . . 15 ⊢ (𝑧 ∈ 𝑣 → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
65	64	adantl 481	. . . . . . . . . . . . . 14 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
66		suprub 12144	. . . . . . . . . . . . . 14 ⊢ (((ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ⊆ ℝ ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ≠ ∅ ∧ ∃𝑑 ∈ ℝ ∀𝑤 ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑤 ≤ 𝑑) ∧ ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))) → ((𝑧𝑀𝑦) + 1) ≤ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
67	58, 59, 62, 65, 66	syl31anc 1375	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → ((𝑧𝑀𝑦) + 1) ≤ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
68		leaddsub 11654	. . . . . . . . . . . . . 14 ⊢ (((𝑧𝑀𝑦) ∈ ℝ ∧ 1 ∈ ℝ ∧ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ) → (((𝑧𝑀𝑦) + 1) ≤ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ↔ (𝑧𝑀𝑦) ≤ (sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) − 1)))
69	18, 55, 57, 68	syl3anc 1373	. . . . . . . . . . . . 13 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → (((𝑧𝑀𝑦) + 1) ≤ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ↔ (𝑧𝑀𝑦) ≤ (sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) − 1)))
70	67, 69	mpbid 232	. . . . . . . . . . . 12 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → (𝑧𝑀𝑦) ≤ (sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) − 1))
71		blss2 24292	. . . . . . . . . . . 12 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (1 ∈ ℝ ∧ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ ∧ (𝑧𝑀𝑦) ≤ (sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) − 1))) → (𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
72	54, 15, 16, 55, 57, 70, 71	syl33anc 1387	. . . . . . . . . . 11 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) → (𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
73	72	ralrimiva 3125	. . . . . . . . . 10 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ∀𝑧 ∈ 𝑣 (𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
74		nfcv 2891	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(𝑥(ball‘𝑀)1)
75		nfcv 2891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧𝑦
76		nfcv 2891	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧(ball‘𝑀)
77		nfmpt1 5206	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑧(𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
78	77	nfrn 5916	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
79		nfcv 2891	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧ℝ
80		nfcv 2891	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑧 <
81	78, 79, 80	nfsup 9402	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑧sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )
82	75, 76, 81	nfov 7417	. . . . . . . . . . . 12 ⊢ Ⅎ𝑧(𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
83	74, 82	nfss 3939	. . . . . . . . . . 11 ⊢ Ⅎ𝑧(𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
84		nfv 1914	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
85		oveq1 7394	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑧 → (𝑥(ball‘𝑀)1) = (𝑧(ball‘𝑀)1))
86	85	sseq1d 3978	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑧 → ((𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ↔ (𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))))
87	83, 84, 86	cbvralw 3280	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ↔ ∀𝑧 ∈ 𝑣 (𝑧(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
88	73, 87	sylibr 234	. . . . . . . . 9 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ∀𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
89		iunss 5009	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ↔ ∀𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
90	88, 89	sylibr 234	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
91	30, 90	eqsstrrd 3982	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑋 ⊆ (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
92	51	rpxrd 12996	. . . . . . . 8 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ*)
93		blssm 24306	. . . . . . . 8 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ*) → (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ⊆ 𝑋)
94	53, 29, 92, 93	syl3anc 1373	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ⊆ 𝑋)
95	91, 94	eqssd 3964	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → 𝑋 = (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
96		oveq2 7395	. . . . . . 7 ⊢ (𝑑 = sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) → (𝑦(ball‘𝑀)𝑑) = (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
97	96	rspceeqv 3611	. . . . . 6 ⊢ ((sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ+ ∧ 𝑋 = (𝑦(ball‘𝑀)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))) → ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑))
98	51, 95, 97	syl2anc 584	. . . . 5 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋)) → ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑))
99	98	rexlimdvaa 3135	. . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑦 ∈ 𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋 → ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
100	99	ralrimdva 3133	. . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋 → ∀𝑦 ∈ 𝑋 ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
101		isbnd 37774	. . . 4 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑦 ∈ 𝑋 ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
102	101	baib 535	. . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → (𝑀 ∈ (Bnd‘𝑋) ↔ ∀𝑦 ∈ 𝑋 ∃𝑑 ∈ ℝ+ 𝑋 = (𝑦(ball‘𝑀)𝑑)))
103	100, 102	sylibrd 259	. 2 ⊢ (𝑀 ∈ (Met‘𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)1) = 𝑋 → 𝑀 ∈ (Bnd‘𝑋)))
104	1, 10, 103	sylc 65	1 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → 𝑀 ∈ (Bnd‘𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3447   ∩ cin 3913   ⊆ wss 3914  ∅c0 4296  𝒫 cpw 4563  ∪ ciun 4955   class class class wbr 5107   ↦ cmpt 5188   Or wor 5545  dom cdm 5638  ran crn 5639  ‘cfv 6511  (class class class)co 7387  Fincfn 8918  supcsup 9391  ℝcr 11067  0cc0 11068  1c1 11069   + caddc 11071  ℝ^*cxr 11207   < clt 11208   ≤ cle 11209   − cmin 11405  ℝ⁺crp 12951  ∞Metcxmet 21249  Metcmet 21250  ballcbl 21251  TotBndctotbnd 37760  Bndcbnd 37761

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145  ax-pre-sup 11146

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-1o 8434  df-er 8671  df-map 8801  df-en 8919  df-dom 8920  df-sdom 8921  df-fin 8922  df-sup 9393  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-div 11836  df-2 12249  df-rp 12952  df-xneg 13072  df-xadd 13073  df-xmul 13074  df-psmet 21256  df-xmet 21257  df-met 21258  df-bl 21259  df-totbnd 37762  df-bnd 37773

This theorem is referenced by:  equivbnd2  37786  prdsbnd2  37789  cntotbnd  37790  cnpwstotbnd  37791