Theorem istotbnd 35927

Description: The predicate "is a totally bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
istotbnd	⊢ (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))

Distinct variable groups:   𝑏,𝑑,𝑣,𝑥,𝑀   𝑋,𝑏,𝑑,𝑣,𝑥

Proof of Theorem istotbnd

Dummy variables 𝑚 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfvex 6807	. 2 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → 𝑋 ∈ V)
2		elfvex 6807	. . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑋 ∈ V)
3	2	adantr 481	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) → 𝑋 ∈ V)
4		fveq2 6774	. . . . . 6 ⊢ (𝑦 = 𝑋 → (Met‘𝑦) = (Met‘𝑋))
5		eqeq2 2750	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → (∪ 𝑣 = 𝑦 ↔ ∪ 𝑣 = 𝑋))
6		rexeq 3343	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → (∃𝑥 ∈ 𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)))
7	6	ralbidv 3112	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → (∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)))
8	5, 7	anbi12d 631	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → ((∪ 𝑣 = 𝑦 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))))
9	8	rexbidv 3226	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (∃𝑣 ∈ Fin (∪ 𝑣 = 𝑦 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))))
10	9	ralbidv 3112	. . . . . 6 ⊢ (𝑦 = 𝑋 → (∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑦 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))))
11	4, 10	rabeqbidv 3420	. . . . 5 ⊢ (𝑦 = 𝑋 → {𝑚 ∈ (Met‘𝑦) ∣ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑦 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑))} = {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))})
12		df-totbnd 35926	. . . . 5 ⊢ TotBnd = (𝑦 ∈ V ↦ {𝑚 ∈ (Met‘𝑦) ∣ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑦 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑))})
13		fvex 6787	. . . . . 6 ⊢ (Met‘𝑋) ∈ V
14	13	rabex 5256	. . . . 5 ⊢ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))} ∈ V
15	11, 12, 14	fvmpt 6875	. . . 4 ⊢ (𝑋 ∈ V → (TotBnd‘𝑋) = {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))})
16	15	eleq2d 2824	. . 3 ⊢ (𝑋 ∈ V → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))}))
17		fveq2 6774	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (ball‘𝑚) = (ball‘𝑀))
18	17	oveqd 7292	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (𝑥(ball‘𝑚)𝑑) = (𝑥(ball‘𝑀)𝑑))
19	18	eqeq2d 2749	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ 𝑏 = (𝑥(ball‘𝑀)𝑑)))
20	19	rexbidv 3226	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
21	20	ralbidv 3112	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
22	21	anbi2d 629	. . . . . 6 ⊢ (𝑚 = 𝑀 → ((∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
23	22	rexbidv 3226	. . . . 5 ⊢ (𝑚 = 𝑀 → (∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
24	23	ralbidv 3112	. . . 4 ⊢ (𝑚 = 𝑀 → (∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)) ↔ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
25	24	elrab 3624	. . 3 ⊢ (𝑀 ∈ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑))} ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
26	16, 25	bitrdi 287	. 2 ⊢ (𝑋 ∈ V → (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))))
27	1, 3, 26	pm5.21nii 380	1 ⊢ (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065  {crab 3068  Vcvv 3432  ∪ cuni 4839  ‘cfv 6433  (class class class)co 7275  Fincfn 8733  ℝ⁺crp 12730  Metcmet 20583  ballcbl 20584  TotBndctotbnd 35924

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  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 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-totbnd 35926

This theorem is referenced by:  istotbnd2  35928  istotbnd3  35929  totbndmet  35930  totbndss  35935  heibor1  35968  heibor  35979