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Theorem istotbnd 36228
Description: The predicate "is a totally bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
istotbnd (𝑀 ∈ (TotBndβ€˜π‘‹) ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
Distinct variable groups:   𝑏,𝑑,𝑣,π‘₯,𝑀   𝑋,𝑏,𝑑,𝑣,π‘₯

Proof of Theorem istotbnd
Dummy variables π‘š 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6880 . 2 (𝑀 ∈ (TotBndβ€˜π‘‹) β†’ 𝑋 ∈ V)
2 elfvex 6880 . . 3 (𝑀 ∈ (Metβ€˜π‘‹) β†’ 𝑋 ∈ V)
32adantr 481 . 2 ((𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))) β†’ 𝑋 ∈ V)
4 fveq2 6842 . . . . . 6 (𝑦 = 𝑋 β†’ (Metβ€˜π‘¦) = (Metβ€˜π‘‹))
5 eqeq2 2748 . . . . . . . . 9 (𝑦 = 𝑋 β†’ (βˆͺ 𝑣 = 𝑦 ↔ βˆͺ 𝑣 = 𝑋))
6 rexeq 3310 . . . . . . . . . 10 (𝑦 = 𝑋 β†’ (βˆƒπ‘₯ ∈ 𝑦 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑) ↔ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑)))
76ralbidv 3174 . . . . . . . . 9 (𝑦 = 𝑋 β†’ (βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑦 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑) ↔ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑)))
85, 7anbi12d 631 . . . . . . . 8 (𝑦 = 𝑋 β†’ ((βˆͺ 𝑣 = 𝑦 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑦 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑)) ↔ (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))))
98rexbidv 3175 . . . . . . 7 (𝑦 = 𝑋 β†’ (βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑦 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑦 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑)) ↔ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))))
109ralbidv 3174 . . . . . 6 (𝑦 = 𝑋 β†’ (βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑦 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑦 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑)) ↔ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))))
114, 10rabeqbidv 3424 . . . . 5 (𝑦 = 𝑋 β†’ {π‘š ∈ (Metβ€˜π‘¦) ∣ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑦 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑦 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))} = {π‘š ∈ (Metβ€˜π‘‹) ∣ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))})
12 df-totbnd 36227 . . . . 5 TotBnd = (𝑦 ∈ V ↦ {π‘š ∈ (Metβ€˜π‘¦) ∣ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑦 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑦 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))})
13 fvex 6855 . . . . . 6 (Metβ€˜π‘‹) ∈ V
1413rabex 5289 . . . . 5 {π‘š ∈ (Metβ€˜π‘‹) ∣ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))} ∈ V
1511, 12, 14fvmpt 6948 . . . 4 (𝑋 ∈ V β†’ (TotBndβ€˜π‘‹) = {π‘š ∈ (Metβ€˜π‘‹) ∣ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))})
1615eleq2d 2823 . . 3 (𝑋 ∈ V β†’ (𝑀 ∈ (TotBndβ€˜π‘‹) ↔ 𝑀 ∈ {π‘š ∈ (Metβ€˜π‘‹) ∣ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))}))
17 fveq2 6842 . . . . . . . . . . 11 (π‘š = 𝑀 β†’ (ballβ€˜π‘š) = (ballβ€˜π‘€))
1817oveqd 7374 . . . . . . . . . 10 (π‘š = 𝑀 β†’ (π‘₯(ballβ€˜π‘š)𝑑) = (π‘₯(ballβ€˜π‘€)𝑑))
1918eqeq2d 2747 . . . . . . . . 9 (π‘š = 𝑀 β†’ (𝑏 = (π‘₯(ballβ€˜π‘š)𝑑) ↔ 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))
2019rexbidv 3175 . . . . . . . 8 (π‘š = 𝑀 β†’ (βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑) ↔ βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))
2120ralbidv 3174 . . . . . . 7 (π‘š = 𝑀 β†’ (βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑) ↔ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))
2221anbi2d 629 . . . . . 6 (π‘š = 𝑀 β†’ ((βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑)) ↔ (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
2322rexbidv 3175 . . . . 5 (π‘š = 𝑀 β†’ (βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑)) ↔ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
2423ralbidv 3174 . . . 4 (π‘š = 𝑀 β†’ (βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑)) ↔ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
2524elrab 3645 . . 3 (𝑀 ∈ {π‘š ∈ (Metβ€˜π‘‹) ∣ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘š)𝑑))} ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑))))
2616, 25bitrdi 286 . 2 (𝑋 ∈ V β†’ (𝑀 ∈ (TotBndβ€˜π‘‹) ↔ (𝑀 ∈ (Metβ€˜π‘‹) ∧ βˆ€π‘‘ ∈ ℝ+ βˆƒπ‘£ ∈ Fin (βˆͺ 𝑣 = 𝑋 ∧ βˆ€π‘ ∈ 𝑣 βˆƒπ‘₯ ∈ 𝑋 𝑏 = (π‘₯(ballβ€˜π‘€)𝑑)))))
271, 3, 26pm5.21nii 379 1 (𝑀 ∈ (TotBndβ€˜π‘‹) ↔ (𝑀 ∈ (Metβ€˜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:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3064  βˆƒwrex 3073  {crab 3407  Vcvv 3445  βˆͺ cuni 4865  β€˜cfv 6496  (class class class)co 7357  Fincfn 8883  β„+crp 12915  Metcmet 20782  ballcbl 20783  TotBndctotbnd 36225
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-iota 6448  df-fun 6498  df-fv 6504  df-ov 7360  df-totbnd 36227
This theorem is referenced by:  istotbnd2  36229  istotbnd3  36230  totbndmet  36231  totbndss  36236  heibor1  36269  heibor  36280
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