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Theorem totbndbnd 36961
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 36941 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 ∈ (TotBndβ€˜π‘‹) β†’ 𝑀 ∈ (Bndβ€˜π‘‹))

Proof of Theorem totbndbnd
Dummy variables 𝑣 𝑑 𝑀 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 totbndmet 36944 . 2 (𝑀 ∈ (TotBndβ€˜π‘‹) β†’ 𝑀 ∈ (Metβ€˜π‘‹))
2 1rp 12983 . . 3 1 ∈ ℝ+
3 istotbnd3 36943 . . . 4 (𝑀 ∈ (TotBndβ€˜π‘‹) ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋))
43simprbi 496 . . 3 (𝑀 ∈ (TotBndβ€˜π‘‹) β†’ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋)
5 oveq2 7420 . . . . . . 7 (𝑑 = 1 β†’ (π‘₯(ballβ€˜π‘€)𝑑) = (π‘₯(ballβ€˜π‘€)1))
65iuneq2d 5027 . . . . . 6 (𝑑 = 1 β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1))
76eqeq1d 2733 . . . . 5 (𝑑 = 1 β†’ (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋 ↔ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋))
87rexbidv 3177 . . . 4 (𝑑 = 1 β†’ (βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋 ↔ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋))
98rspcv 3609 . . 3 (1 ∈ ℝ+ β†’ (βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) = 𝑋 β†’ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋))
102, 4, 9mpsyl 68 . 2 (𝑀 ∈ (TotBndβ€˜π‘‹) β†’ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)
11 simplll 772 . . . . . . . . . . 11 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ 𝑀 ∈ (Metβ€˜π‘‹))
12 elfpw 9357 . . . . . . . . . . . . . 14 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 βŠ† 𝑋 ∧ 𝑣 ∈ Fin))
1312simplbi 497 . . . . . . . . . . . . 13 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) β†’ 𝑣 βŠ† 𝑋)
1413ad2antrl 725 . . . . . . . . . . . 12 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ 𝑣 βŠ† 𝑋)
1514sselda 3983 . . . . . . . . . . 11 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ 𝑧 ∈ 𝑋)
16 simpllr 773 . . . . . . . . . . 11 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ 𝑦 ∈ 𝑋)
17 metcl 24059 . . . . . . . . . . 11 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ (𝑧𝑀𝑦) ∈ ℝ)
1811, 15, 16, 17syl3anc 1370 . . . . . . . . . 10 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ (𝑧𝑀𝑦) ∈ ℝ)
19 metge0 24072 . . . . . . . . . . 11 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) β†’ 0 ≀ (𝑧𝑀𝑦))
2011, 15, 16, 19syl3anc 1370 . . . . . . . . . 10 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ 0 ≀ (𝑧𝑀𝑦))
2118, 20ge0p1rpd 13051 . . . . . . . . 9 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ ((𝑧𝑀𝑦) + 1) ∈ ℝ+)
2221fmpttd 7117 . . . . . . . 8 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)):π‘£βŸΆβ„+)
2322frnd 6726 . . . . . . 7 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) βŠ† ℝ+)
2412simprbi 496 . . . . . . . . . 10 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) β†’ 𝑣 ∈ Fin)
25 mptfi 9354 . . . . . . . . . 10 (𝑣 ∈ Fin β†’ (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
26 rnfi 9338 . . . . . . . . . 10 ((𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
2724, 25, 263syl 18 . . . . . . . . 9 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
2827ad2antrl 725 . . . . . . . 8 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
29 simplr 766 . . . . . . . . . 10 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ 𝑦 ∈ 𝑋)
30 simprr 770 . . . . . . . . . 10 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)
3129, 30eleqtrrd 2835 . . . . . . . . 9 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ 𝑦 ∈ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1))
32 ne0i 4335 . . . . . . . . 9 (𝑦 ∈ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) β‰  βˆ…)
33 dm0rn0 5925 . . . . . . . . . . 11 (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = βˆ… ↔ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = βˆ…)
34 ovex 7445 . . . . . . . . . . . . . . 15 ((𝑧𝑀𝑦) + 1) ∈ V
35 eqid 2731 . . . . . . . . . . . . . . 15 (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
3634, 35dmmpti 6695 . . . . . . . . . . . . . 14 dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = 𝑣
3736eqeq1i 2736 . . . . . . . . . . . . 13 (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = βˆ… ↔ 𝑣 = βˆ…)
38 iuneq1 5014 . . . . . . . . . . . . 13 (𝑣 = βˆ… β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = βˆͺ π‘₯ ∈ βˆ… (π‘₯(ballβ€˜π‘€)1))
3937, 38sylbi 216 . . . . . . . . . . . 12 (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = βˆ… β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = βˆͺ π‘₯ ∈ βˆ… (π‘₯(ballβ€˜π‘€)1))
40 0iun 5067 . . . . . . . . . . . 12 βˆͺ π‘₯ ∈ βˆ… (π‘₯(ballβ€˜π‘€)1) = βˆ…
4139, 40eqtrdi 2787 . . . . . . . . . . 11 (dom (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = βˆ… β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = βˆ…)
4233, 41sylbir 234 . . . . . . . . . 10 (ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) = βˆ… β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = βˆ…)
4342necon3i 2972 . . . . . . . . 9 (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) β‰  βˆ… β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) β‰  βˆ…)
4431, 32, 433syl 18 . . . . . . . 8 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) β‰  βˆ…)
45 rpssre 12986 . . . . . . . . 9 ℝ+ βŠ† ℝ
4623, 45sstrdi 3995 . . . . . . . 8 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) βŠ† ℝ)
47 ltso 11299 . . . . . . . . 9 < Or ℝ
48 fisupcl 9467 . . . . . . . . 9 (( < Or ℝ ∧ (ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) β‰  βˆ… ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) βŠ† ℝ)) β†’ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
4947, 48mpan 687 . . . . . . . 8 ((ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) β‰  βˆ… ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) βŠ† ℝ) β†’ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
5028, 44, 46, 49syl3anc 1370 . . . . . . 7 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
5123, 50sseldd 3984 . . . . . 6 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ+)
52 metxmet 24061 . . . . . . . . . . . . . 14 (𝑀 ∈ (Metβ€˜π‘‹) β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))
5352ad2antrr 723 . . . . . . . . . . . . 13 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))
5453adantr 480 . . . . . . . . . . . 12 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ 𝑀 ∈ (∞Metβ€˜π‘‹))
55 1red 11220 . . . . . . . . . . . 12 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ 1 ∈ ℝ)
5646, 50sseldd 3984 . . . . . . . . . . . . 13 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ)
5756adantr 480 . . . . . . . . . . . 12 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ)
5846adantr 480 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) βŠ† ℝ)
5944adantr 480 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) β‰  βˆ…)
6028adantr 480 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin)
61 fimaxre2 12164 . . . . . . . . . . . . . . 15 ((ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) βŠ† ℝ ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) ∈ Fin) β†’ βˆƒπ‘‘ ∈ ℝ βˆ€π‘€ ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑀 ≀ 𝑑)
6258, 60, 61syl2anc 583 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ βˆƒπ‘‘ ∈ ℝ βˆ€π‘€ ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑀 ≀ 𝑑)
6335elrnmpt1 5958 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ 𝑣 ∧ ((𝑧𝑀𝑦) + 1) ∈ V) β†’ ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
6434, 63mpan2 688 . . . . . . . . . . . . . . 15 (𝑧 ∈ 𝑣 β†’ ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
6564adantl 481 . . . . . . . . . . . . . 14 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)))
66 suprub 12180 . . . . . . . . . . . . . 14 (((ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) βŠ† ℝ ∧ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)) β‰  βˆ… ∧ βˆƒπ‘‘ ∈ ℝ βˆ€π‘€ ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))𝑀 ≀ 𝑑) ∧ ((𝑧𝑀𝑦) + 1) ∈ ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))) β†’ ((𝑧𝑀𝑦) + 1) ≀ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
6758, 59, 62, 65, 66syl31anc 1372 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ ((𝑧𝑀𝑦) + 1) ≀ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
68 leaddsub 11695 . . . . . . . . . . . . . 14 (((𝑧𝑀𝑦) ∈ ℝ ∧ 1 ∈ ℝ ∧ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ) β†’ (((𝑧𝑀𝑦) + 1) ≀ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ↔ (𝑧𝑀𝑦) ≀ (sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) βˆ’ 1)))
6918, 55, 57, 68syl3anc 1370 . . . . . . . . . . . . 13 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ (((𝑧𝑀𝑦) + 1) ≀ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ↔ (𝑧𝑀𝑦) ≀ (sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) βˆ’ 1)))
7067, 69mpbid 231 . . . . . . . . . . . 12 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ (𝑧𝑀𝑦) ≀ (sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) βˆ’ 1))
71 blss2 24131 . . . . . . . . . . . 12 (((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑧 ∈ 𝑋 ∧ 𝑦 ∈ 𝑋) ∧ (1 ∈ ℝ ∧ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ ∧ (𝑧𝑀𝑦) ≀ (sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) βˆ’ 1))) β†’ (𝑧(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
7254, 15, 16, 55, 57, 70, 71syl33anc 1384 . . . . . . . . . . 11 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) ∧ 𝑧 ∈ 𝑣) β†’ (𝑧(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
7372ralrimiva 3145 . . . . . . . . . 10 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ βˆ€π‘§ ∈ 𝑣 (𝑧(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
74 nfcv 2902 . . . . . . . . . . . 12 Ⅎ𝑧(π‘₯(ballβ€˜π‘€)1)
75 nfcv 2902 . . . . . . . . . . . . 13 Ⅎ𝑧𝑦
76 nfcv 2902 . . . . . . . . . . . . 13 Ⅎ𝑧(ballβ€˜π‘€)
77 nfmpt1 5257 . . . . . . . . . . . . . . 15 Ⅎ𝑧(𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
7877nfrn 5952 . . . . . . . . . . . . . 14 Ⅎ𝑧ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1))
79 nfcv 2902 . . . . . . . . . . . . . 14 Ⅎ𝑧ℝ
80 nfcv 2902 . . . . . . . . . . . . . 14 Ⅎ𝑧 <
8178, 79, 80nfsup 9449 . . . . . . . . . . . . 13 Ⅎ𝑧sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )
8275, 76, 81nfov 7442 . . . . . . . . . . . 12 Ⅎ𝑧(𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
8374, 82nfss 3975 . . . . . . . . . . 11 Ⅎ𝑧(π‘₯(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
84 nfv 1916 . . . . . . . . . . 11 β„²π‘₯(𝑧(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))
85 oveq1 7419 . . . . . . . . . . . 12 (π‘₯ = 𝑧 β†’ (π‘₯(ballβ€˜π‘€)1) = (𝑧(ballβ€˜π‘€)1))
8685sseq1d 4014 . . . . . . . . . . 11 (π‘₯ = 𝑧 β†’ ((π‘₯(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ↔ (𝑧(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))))
8783, 84, 86cbvralw 3302 . . . . . . . . . 10 (βˆ€π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ↔ βˆ€π‘§ ∈ 𝑣 (𝑧(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
8873, 87sylibr 233 . . . . . . . . 9 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ βˆ€π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
89 iunss 5049 . . . . . . . . 9 (βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) ↔ βˆ€π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
9088, 89sylibr 233 . . . . . . . 8 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
9130, 90eqsstrrd 4022 . . . . . . 7 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ 𝑋 βŠ† (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
9251rpxrd 13022 . . . . . . . 8 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ*)
93 blssm 24145 . . . . . . . 8 ((𝑀 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ*) β†’ (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) βŠ† 𝑋)
9453, 29, 92, 93syl3anc 1370 . . . . . . 7 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )) βŠ† 𝑋)
9591, 94eqssd 4000 . . . . . 6 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ 𝑋 = (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
96 oveq2 7420 . . . . . . 7 (𝑑 = sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) β†’ (𝑦(ballβ€˜π‘€)𝑑) = (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < )))
9796rspceeqv 3634 . . . . . 6 ((sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ) ∈ ℝ+ ∧ 𝑋 = (𝑦(ballβ€˜π‘€)sup(ran (𝑧 ∈ 𝑣 ↦ ((𝑧𝑀𝑦) + 1)), ℝ, < ))) β†’ βˆƒπ‘‘ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)𝑑))
9851, 95, 97syl2anc 583 . . . . 5 (((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋)) β†’ βˆƒπ‘‘ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)𝑑))
9998rexlimdvaa 3155 . . . 4 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ (βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋 β†’ βˆƒπ‘‘ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)𝑑)))
10099ralrimdva 3153 . . 3 (𝑀 ∈ (Metβ€˜π‘‹) β†’ (βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋 β†’ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘‘ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)𝑑)))
101 isbnd 36952 . . . 4 (𝑀 ∈ (Bndβ€˜π‘‹) ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘‘ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)𝑑)))
102101baib 535 . . 3 (𝑀 ∈ (Metβ€˜π‘‹) β†’ (𝑀 ∈ (Bndβ€˜π‘‹) ↔ βˆ€π‘¦ ∈ 𝑋 βˆƒπ‘‘ ∈ ℝ+ 𝑋 = (𝑦(ballβ€˜π‘€)𝑑)))
103100, 102sylibrd 258 . 2 (𝑀 ∈ (Metβ€˜π‘‹) β†’ (βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)1) = 𝑋 β†’ 𝑀 ∈ (Bndβ€˜π‘‹)))
1041, 10, 103sylc 65 1 (𝑀 ∈ (TotBndβ€˜π‘‹) β†’ 𝑀 ∈ (Bndβ€˜π‘‹))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105   β‰  wne 2939  βˆ€wral 3060  βˆƒwrex 3069  Vcvv 3473   ∩ cin 3948   βŠ† wss 3949  βˆ…c0 4323  π’« cpw 4603  βˆͺ ciun 4998   class class class wbr 5149   ↦ cmpt 5232   Or wor 5588  dom cdm 5677  ran crn 5678  β€˜cfv 6544  (class class class)co 7412  Fincfn 8942  supcsup 9438  β„cr 11112  0cc0 11113  1c1 11114   + caddc 11116  β„*cxr 11252   < clt 11253   ≀ cle 11254   βˆ’ cmin 11449  β„+crp 12979  βˆžMetcxmet 21130  Metcmet 21131  ballcbl 21132  TotBndctotbnd 36938  Bndcbnd 36939
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169  ax-resscn 11170  ax-1cn 11171  ax-icn 11172  ax-addcl 11173  ax-addrcl 11174  ax-mulcl 11175  ax-mulrcl 11176  ax-mulcom 11177  ax-addass 11178  ax-mulass 11179  ax-distr 11180  ax-i2m1 11181  ax-1ne0 11182  ax-1rid 11183  ax-rnegex 11184  ax-rrecex 11185  ax-cnre 11186  ax-pre-lttri 11187  ax-pre-lttrn 11188  ax-pre-ltadd 11189  ax-pre-mulgt0 11190  ax-pre-sup 11191
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3375  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-1st 7978  df-2nd 7979  df-1o 8469  df-er 8706  df-map 8825  df-en 8943  df-dom 8944  df-sdom 8945  df-fin 8946  df-sup 9440  df-pnf 11255  df-mnf 11256  df-xr 11257  df-ltxr 11258  df-le 11259  df-sub 11451  df-neg 11452  df-div 11877  df-2 12280  df-rp 12980  df-xneg 13097  df-xadd 13098  df-xmul 13099  df-psmet 21137  df-xmet 21138  df-met 21139  df-bl 21140  df-totbnd 36940  df-bnd 36951
This theorem is referenced by:  equivbnd2  36964  prdsbnd2  36967  cntotbnd  36968  cnpwstotbnd  36969
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