Theorem istotbnd 38090

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 6875	. 2 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → 𝑋 ∈ V)
2		elfvex 6875	. . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑋 ∈ V)
3	2	adantr 480	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) → 𝑋 ∈ V)
4		fveq2 6840	. . . . . 6 ⊢ (𝑦 = 𝑋 → (Met‘𝑦) = (Met‘𝑋))
5		eqeq2 2748	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → (∪ 𝑣 = 𝑦 ↔ ∪ 𝑣 = 𝑋))
6		rexeq 3291	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → (∃𝑥 ∈ 𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)))
7	6	ralbidv 3160	. . . . . . . . 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 633	. . . . . . . 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 3161	. . . . . . 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 3160	. . . . . 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 3407	. . . . 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 38089	. . . . 5 ⊢ TotBnd = (𝑦 ∈ V ↦ {𝑚 ∈ (Met‘𝑦) ∣ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑦 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑))})
13		fvex 6853	. . . . . 6 ⊢ (Met‘𝑋) ∈ V
14	13	rabex 5280	. . . . 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 6947	. . . 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 2822	. . 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 6840	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (ball‘𝑚) = (ball‘𝑀))
18	17	oveqd 7384	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (𝑥(ball‘𝑚)𝑑) = (𝑥(ball‘𝑀)𝑑))
19	18	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ 𝑏 = (𝑥(ball‘𝑀)𝑑)))
20	19	rexbidv 3161	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
21	20	ralbidv 3160	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
22	21	anbi2d 631	. . . . . 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 3161	. . . . 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 3160	. . . 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 3634	. . 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 378	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 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  ∃wrex 3061  {crab 3389  Vcvv 3429  ∪ cuni 4850  ‘cfv 6498  (class class class)co 7367  Fincfn 8893  ℝ⁺crp 12942  Metcmet 21338  ballcbl 21339  TotBndctotbnd 38087

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-totbnd 38089

This theorem is referenced by:  istotbnd2  38091  istotbnd3  38092  totbndmet  38093  totbndss  38098  heibor1  38131  heibor  38142