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Theorem sstotbnd3 37157
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 37155 . . 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 3959 . . . . . . . . 9 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 ∈ 𝒫 𝑋 ∧ 𝑣 ∈ Fin))
4 rabfi 9271 . . . . . . . . . 10 (𝑣 ∈ Fin β†’ {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} ∈ Fin)
54anim2i 616 . . . . . . . . 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 616 . . . . . . 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 458 . . . . . 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 642 . . . . . 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 3073 . . . 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 3077 . . 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, 12syl6bi 253 . 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 4072 . . . . . . . 8 {π‘₯ ∈ 𝑣 ∣ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…} βŠ† 𝑣
15 elpwi 4604 . . . . . . . . 9 (𝑣 ∈ 𝒫 𝑋 β†’ 𝑣 βŠ† 𝑋)
1615ad2antlr 724 . . . . . . . 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 3988 . . . . . . 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 770 . . . . . . 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 9356 . . . . . . 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 582 . . . . . 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 3972 . . . . . . . . . . . 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 4994 . . . . . . . . . . . 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 4459 . . . . . . . . . . . . . . 15 ((𝑧 ∈ (π‘₯(ballβ€˜π‘€)𝑑) ∧ 𝑧 ∈ π‘Œ) β†’ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…)
2524expcom 413 . . . . . . . . . . . . . 14 (𝑧 ∈ π‘Œ β†’ (𝑧 ∈ (π‘₯(ballβ€˜π‘€)𝑑) β†’ ((π‘₯(ballβ€˜π‘€)𝑑) ∩ π‘Œ) β‰  βˆ…))
2625ancrd 551 . . . . . . . . . . . . 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 3164 . . . . . . . . . . . 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 407 . . . . . . . . . . 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 587 . . . . . . . . . 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 4994 . . . . . . . . . . 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 7412 . . . . . . . . . . . . 13 (𝑦 = π‘₯ β†’ (𝑦(ballβ€˜π‘€)𝑑) = (π‘₯(ballβ€˜π‘€)𝑑))
3231eleq2d 2813 . . . . . . . . . . . 12 (𝑦 = π‘₯ β†’ (𝑧 ∈ (𝑦(ballβ€˜π‘€)𝑑) ↔ 𝑧 ∈ (π‘₯(ballβ€˜π‘€)𝑑)))
3332rexrab2 3691 . . . . . . . . . . 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 275 . . . . . . . . . 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 412 . . . . . . . 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 3983 . . . . . . 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 725 . . . . . 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 5006 . . . . . . . 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 4009 . . . . . . 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 3606 . . . . . 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 583 . . . . 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 3151 . . . 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 3163 . . 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 37155 . . 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 259 . 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 395   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆ€wral 3055  βˆƒwrex 3064  {crab 3426   ∩ cin 3942   βŠ† wss 3943  βˆ…c0 4317  π’« cpw 4597  βˆͺ ciun 4990   Γ— cxp 5667   β†Ύ cres 5671  β€˜cfv 6537  (class class class)co 7405  Fincfn 8941  β„+crp 12980  Metcmet 21226  ballcbl 21227  TotBndctotbnd 37147
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-1o 8467  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-2 12279  df-rp 12981  df-xneg 13098  df-xadd 13099  df-xmul 13100  df-psmet 21232  df-xmet 21233  df-met 21234  df-bl 21235  df-totbnd 37149
This theorem is referenced by:  cntotbnd  37177
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