Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sstotbnd3 Structured version   Visualization version   GIF version

Theorem sstotbnd3 37306
Description: Use a net that is not necessarily finite, but for which only finitely many balls meet the subset. (Contributed by Mario Carneiro, 14-Sep-2015.)
Hypothesis
Ref Expression
sstotbnd.2 𝑁 = (𝑀 β†Ύ (π‘Œ Γ— π‘Œ))
Assertion
Ref Expression
sstotbnd3 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝑁 ∈ (TotBndβ€˜π‘Œ) ↔ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ 𝒫 𝑋(π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin)))
Distinct variable groups:   𝑣,𝑑,π‘₯,𝑀   𝑋,𝑑,𝑣,π‘₯   𝑁,𝑑,𝑣,π‘₯   π‘Œ,𝑑,𝑣,π‘₯

Proof of Theorem sstotbnd3
Dummy variables 𝑀 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sstotbnd.2 . . . 4 𝑁 = (𝑀 β†Ύ (π‘Œ Γ— π‘Œ))
21sstotbnd2 37304 . . 3 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝑁 ∈ (TotBndβ€˜π‘Œ) ↔ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑)))
3 elin 3955 . . . . . . . . 9 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 ∈ 𝒫 𝑋 ∧ 𝑣 ∈ Fin))
4 rabfi 9292 . . . . . . . . . 10 (𝑣 ∈ Fin β†’ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin)
54anim2i 615 . . . . . . . . 9 ((𝑣 ∈ 𝒫 𝑋 ∧ 𝑣 ∈ Fin) β†’ (𝑣 ∈ 𝒫 𝑋 ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin))
63, 5sylbi 216 . . . . . . . 8 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) β†’ (𝑣 ∈ 𝒫 𝑋 ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin))
76anim2i 615 . . . . . . 7 ((π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) β†’ (π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ (𝑣 ∈ 𝒫 𝑋 ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin)))
87ancoms 457 . . . . . 6 ((𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑)) β†’ (π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ (𝑣 ∈ 𝒫 𝑋 ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin)))
9 an12 643 . . . . . 6 ((π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ (𝑣 ∈ 𝒫 𝑋 ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin)) ↔ (𝑣 ∈ 𝒫 𝑋 ∧ (π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin)))
108, 9sylib 217 . . . . 5 ((𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑)) β†’ (𝑣 ∈ 𝒫 𝑋 ∧ (π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin)))
1110reximi2 3069 . . . 4 (βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) β†’ βˆƒπ‘£ ∈ 𝒫 𝑋(π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin))
1211ralimi 3073 . . 3 (βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ (𝒫 𝑋 ∩ Fin)π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) β†’ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ 𝒫 𝑋(π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin))
132, 12biimtrdi 252 . 2 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝑁 ∈ (TotBndβ€˜π‘Œ) β†’ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ 𝒫 𝑋(π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin)))
14 ssrab2 4069 . . . . . . . 8 {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} βŠ† 𝑣
15 elpwi 4605 . . . . . . . . 9 (𝑣 ∈ 𝒫 𝑋 β†’ 𝑣 βŠ† 𝑋)
1615ad2antlr 725 . . . . . . . 8 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin)) β†’ 𝑣 βŠ† 𝑋)
1714, 16sstrid 3984 . . . . . . 7 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin)) β†’ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} βŠ† 𝑋)
18 simprr 771 . . . . . . 7 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin)) β†’ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin)
19 elfpw 9378 . . . . . . 7 ({π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ (𝒫 𝑋 ∩ Fin) ↔ ({π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} βŠ† 𝑋 ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin))
2017, 18, 19sylanbrc 581 . . . . . 6 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin)) β†’ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ (𝒫 𝑋 ∩ Fin))
21 ssel2 3967 . . . . . . . . . . . 12 ((π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ 𝑧 ∈ π‘Œ) β†’ 𝑧 ∈ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑))
22 eliun 4995 . . . . . . . . . . . 12 (𝑧 ∈ βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ↔ βˆƒπ‘₯ ∈ 𝑣 𝑧 ∈ (π‘₯(ballβ€˜π‘€)𝑑))
2321, 22sylib 217 . . . . . . . . . . 11 ((π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ 𝑧 ∈ π‘Œ) β†’ βˆƒπ‘₯ ∈ 𝑣 𝑧 ∈ (π‘₯(ballβ€˜π‘€)𝑑))
24 inelcm 4460 . . . . . . . . . . . . . . 15 ((𝑧 ∈ (π‘₯(ballβ€˜π‘€)𝑑) ∧ 𝑧 ∈ π‘Œ) β†’ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…)
2524expcom 412 . . . . . . . . . . . . . 14 (𝑧 ∈ π‘Œ β†’ (𝑧 ∈ (π‘₯(ballβ€˜π‘€)𝑑) β†’ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…))
2625ancrd 550 . . . . . . . . . . . . 13 (𝑧 ∈ π‘Œ β†’ (𝑧 ∈ (π‘₯(ballβ€˜π‘€)𝑑) β†’ (((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ… ∧ 𝑧 ∈ (π‘₯(ballβ€˜π‘€)𝑑))))
2726reximdv 3160 . . . . . . . . . . . 12 (𝑧 ∈ π‘Œ β†’ (βˆƒπ‘₯ ∈ 𝑣 𝑧 ∈ (π‘₯(ballβ€˜π‘€)𝑑) β†’ βˆƒπ‘₯ ∈ 𝑣 (((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ… ∧ 𝑧 ∈ (π‘₯(ballβ€˜π‘€)𝑑))))
2827impcom 406 . . . . . . . . . . 11 ((βˆƒπ‘₯ ∈ 𝑣 𝑧 ∈ (π‘₯(ballβ€˜π‘€)𝑑) ∧ 𝑧 ∈ π‘Œ) β†’ βˆƒπ‘₯ ∈ 𝑣 (((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ… ∧ 𝑧 ∈ (π‘₯(ballβ€˜π‘€)𝑑)))
2923, 28sylancom 586 . . . . . . . . . 10 ((π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ 𝑧 ∈ π‘Œ) β†’ βˆƒπ‘₯ ∈ 𝑣 (((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ… ∧ 𝑧 ∈ (π‘₯(ballβ€˜π‘€)𝑑)))
30 eliun 4995 . . . . . . . . . . 11 (𝑧 ∈ βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} (𝑦(ballβ€˜π‘€)𝑑) ↔ βˆƒπ‘¦ ∈ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…}𝑧 ∈ (𝑦(ballβ€˜π‘€)𝑑))
31 oveq1 7423 . . . . . . . . . . . . 13 (𝑦 = π‘₯ β†’ (𝑦(ballβ€˜π‘€)𝑑) = (π‘₯(ballβ€˜π‘€)𝑑))
3231eleq2d 2811 . . . . . . . . . . . 12 (𝑦 = π‘₯ β†’ (𝑧 ∈ (𝑦(ballβ€˜π‘€)𝑑) ↔ 𝑧 ∈ (π‘₯(ballβ€˜π‘€)𝑑)))
3332rexrab2 3687 . . . . . . . . . . 11 (βˆƒπ‘¦ ∈ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…}𝑧 ∈ (𝑦(ballβ€˜π‘€)𝑑) ↔ βˆƒπ‘₯ ∈ 𝑣 (((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ… ∧ 𝑧 ∈ (π‘₯(ballβ€˜π‘€)𝑑)))
3430, 33bitri 274 . . . . . . . . . 10 (𝑧 ∈ βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} (𝑦(ballβ€˜π‘€)𝑑) ↔ βˆƒπ‘₯ ∈ 𝑣 (((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ… ∧ 𝑧 ∈ (π‘₯(ballβ€˜π‘€)𝑑)))
3529, 34sylibr 233 . . . . . . . . 9 ((π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ 𝑧 ∈ π‘Œ) β†’ 𝑧 ∈ βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} (𝑦(ballβ€˜π‘€)𝑑))
3635ex 411 . . . . . . . 8 (π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) β†’ (𝑧 ∈ π‘Œ β†’ 𝑧 ∈ βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} (𝑦(ballβ€˜π‘€)𝑑)))
3736ssrdv 3978 . . . . . . 7 (π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) β†’ π‘Œ βŠ† βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} (𝑦(ballβ€˜π‘€)𝑑))
3837ad2antrl 726 . . . . . 6 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin)) β†’ π‘Œ βŠ† βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} (𝑦(ballβ€˜π‘€)𝑑))
39 iuneq1 5007 . . . . . . . 8 (𝑀 = {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} β†’ βˆͺ 𝑦 ∈ 𝑀 (𝑦(ballβ€˜π‘€)𝑑) = βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} (𝑦(ballβ€˜π‘€)𝑑))
4039sseq2d 4005 . . . . . . 7 (𝑀 = {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} β†’ (π‘Œ βŠ† βˆͺ 𝑦 ∈ 𝑀 (𝑦(ballβ€˜π‘€)𝑑) ↔ π‘Œ βŠ† βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} (𝑦(ballβ€˜π‘€)𝑑)))
4140rspcev 3601 . . . . . 6 (({π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ (𝒫 𝑋 ∩ Fin) ∧ π‘Œ βŠ† βˆͺ 𝑦 ∈ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} (𝑦(ballβ€˜π‘€)𝑑)) β†’ βˆƒπ‘€ ∈ (𝒫 𝑋 ∩ Fin)π‘Œ βŠ† βˆͺ 𝑦 ∈ 𝑀 (𝑦(ballβ€˜π‘€)𝑑))
4220, 38, 41syl2anc 582 . . . . 5 ((((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin)) β†’ βˆƒπ‘€ ∈ (𝒫 𝑋 ∩ Fin)π‘Œ βŠ† βˆͺ 𝑦 ∈ 𝑀 (𝑦(ballβ€˜π‘€)𝑑))
4342rexlimdva2 3147 . . . 4 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (βˆƒπ‘£ ∈ 𝒫 𝑋(π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin) β†’ βˆƒπ‘€ ∈ (𝒫 𝑋 ∩ Fin)π‘Œ βŠ† βˆͺ 𝑦 ∈ 𝑀 (𝑦(ballβ€˜π‘€)𝑑)))
4443ralimdv 3159 . . 3 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ 𝒫 𝑋(π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin) β†’ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘€ ∈ (𝒫 𝑋 ∩ Fin)π‘Œ βŠ† βˆͺ 𝑦 ∈ 𝑀 (𝑦(ballβ€˜π‘€)𝑑)))
451sstotbnd2 37304 . . 3 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝑁 ∈ (TotBndβ€˜π‘Œ) ↔ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘€ ∈ (𝒫 𝑋 ∩ Fin)π‘Œ βŠ† βˆͺ 𝑦 ∈ 𝑀 (𝑦(ballβ€˜π‘€)𝑑)))
4644, 45sylibrd 258 . 2 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ 𝒫 𝑋(π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin) β†’ 𝑁 ∈ (TotBndβ€˜π‘Œ)))
4713, 46impbid 211 1 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝑁 ∈ (TotBndβ€˜π‘Œ) ↔ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ 𝒫 𝑋(π‘Œ βŠ† βˆͺ π‘₯ ∈ 𝑣 (π‘₯(ballβ€˜π‘€)𝑑) ∧ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆ€wral 3051  βˆƒwrex 3060  {crab 3419   ∩ cin 3938   βŠ† wss 3939  βˆ…c0 4318  π’« cpw 4598  βˆͺ ciun 4991   Γ— cxp 5670   β†Ύ cres 5674  β€˜cfv 6543  (class class class)co 7416  Fincfn 8962  β„+crp 13006  Metcmet 21269  ballcbl 21270  TotBndctotbnd 37296
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7738  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-pss 3959  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7372  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7869  df-1st 7991  df-2nd 7992  df-1o 8485  df-er 8723  df-map 8845  df-en 8963  df-dom 8964  df-sdom 8965  df-fin 8966  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-div 11902  df-2 12305  df-rp 13007  df-xneg 13124  df-xadd 13125  df-xmul 13126  df-psmet 21275  df-xmet 21276  df-met 21277  df-bl 21278  df-totbnd 37298
This theorem is referenced by:  cntotbnd  37326
  Copyright terms: Public domain W3C validator