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Theorem istotbnd2 37151
Description: The predicate "is a totally bounded metric space." (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
istotbnd2 (𝑀 ∈ (Metβ€˜π‘‹) β†’ (𝑀 ∈ (TotBndβ€˜π‘‹) ↔ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
Distinct variable groups:   𝑀,𝑑,𝑣,𝑏,π‘₯   𝑋,𝑑,𝑣,𝑏,π‘₯

Proof of Theorem istotbnd2
StepHypRef Expression
1 istotbnd 37150 . 2 (𝑀 ∈ (TotBndβ€˜π‘‹) ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
21baib 535 1 (𝑀 ∈ (Metβ€˜π‘‹) β†’ (𝑀 ∈ (TotBndβ€˜π‘‹) ↔ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  βˆƒwrex 3064  βˆͺ cuni 4902  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-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-ov 7408  df-totbnd 37149
This theorem is referenced by: (None)
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