Theorem istotbnd 38401

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 6918	. 2 ⊢ (𝑀 ∈ (TotBnd‘𝑋) → 𝑋 ∈ V)
2		elfvex 6918	. . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑋 ∈ V)
3	2	adantr 485	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑋 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) → 𝑋 ∈ V)
4		fveq2 6883	. . . . . 6 ⊢ (𝑦 = 𝑋 → (Met‘𝑦) = (Met‘𝑋))
5		eqeq2 2775	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → (∪ 𝑣 = 𝑦 ↔ ∪ 𝑣 = 𝑋))
6		rexeq 3319	. . . . . . . . . 10 ⊢ (𝑦 = 𝑋 → (∃𝑥 ∈ 𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑)))
7	6	ralbidv 3188	. . . . . . . . 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 643	. . . . . . . 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 3189	. . . . . . 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 3188	. . . . . 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 3434	. . . . 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 38400	. . . . 5 ⊢ TotBnd = (𝑦 ∈ V ↦ {𝑚 ∈ (Met‘𝑦) ∣ ∀𝑑 ∈ ℝ+ ∃𝑣 ∈ Fin (∪ 𝑣 = 𝑦 ∧ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑦 𝑏 = (𝑥(ball‘𝑚)𝑑))})
13		fvex 6896	. . . . . 6 ⊢ (Met‘𝑋) ∈ V
14	13	rabex 5311	. . . . 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 6991	. . . 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 2849	. . 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 6883	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑀 → (ball‘𝑚) = (ball‘𝑀))
18	17	oveqd 7429	. . . . . . . . . 10 ⊢ (𝑚 = 𝑀 → (𝑥(ball‘𝑚)𝑑) = (𝑥(ball‘𝑀)𝑑))
19	18	eqeq2d 2774	. . . . . . . . 9 ⊢ (𝑚 = 𝑀 → (𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ 𝑏 = (𝑥(ball‘𝑀)𝑑)))
20	19	rexbidv 3189	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
21	20	ralbidv 3188	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑚)𝑑) ↔ ∀𝑏 ∈ 𝑣 ∃𝑥 ∈ 𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
22	21	anbi2d 641	. . . . . 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 3189	. . . . 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 3188	. . . 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 3651	. . 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 290	. 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 381	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 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  {crab 3416  Vcvv 3455  ∪ cuni 4873  ‘cfv 6538  (class class class)co 7412  Fincfn 8944  ℝ⁺crp 13017  Metcmet 21489  ballcbl 21490  TotBndctotbnd 38398

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7415  df-totbnd 38400

This theorem is referenced by:  istotbnd2  38402  istotbnd3  38403  totbndmet  38404  totbndss  38409  heibor1  38442  heibor  38453