Theorem isbnd 36953

Description: The predicate "is a bounded metric space". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 12-Sep-2015.)

Assertion

Ref	Expression
isbnd	⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))

Distinct variable groups:   𝑥,𝑟,𝑀   𝑋,𝑟,𝑥

Proof of Theorem isbnd

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

Step	Hyp	Ref	Expression
1		elfvex 6930	. 2 ⊢ (𝑀 ∈ (Bnd‘𝑋) → 𝑋 ∈ V)
2		elfvex 6930	. . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑋 ∈ V)
3	2	adantr 479	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑋 ∈ V)
4		fveq2 6892	. . . . . 6 ⊢ (𝑦 = 𝑋 → (Met‘𝑦) = (Met‘𝑋))
5		eqeq1 2734	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑦 = (𝑥(ball‘𝑚)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑚)𝑟)))
6	5	rexbidv 3176	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (∃𝑟 ∈ ℝ+ 𝑦 = (𝑥(ball‘𝑚)𝑟) ↔ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)))
7	6	raleqbi1dv 3331	. . . . . 6 ⊢ (𝑦 = 𝑋 → (∀𝑥 ∈ 𝑦 ∃𝑟 ∈ ℝ+ 𝑦 = (𝑥(ball‘𝑚)𝑟) ↔ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)))
8	4, 7	rabeqbidv 3447	. . . . 5 ⊢ (𝑦 = 𝑋 → {𝑚 ∈ (Met‘𝑦) ∣ ∀𝑥 ∈ 𝑦 ∃𝑟 ∈ ℝ+ 𝑦 = (𝑥(ball‘𝑚)𝑟)} = {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)})
9		df-bnd 36952	. . . . 5 ⊢ Bnd = (𝑦 ∈ V ↦ {𝑚 ∈ (Met‘𝑦) ∣ ∀𝑥 ∈ 𝑦 ∃𝑟 ∈ ℝ+ 𝑦 = (𝑥(ball‘𝑚)𝑟)})
10		fvex 6905	. . . . . 6 ⊢ (Met‘𝑋) ∈ V
11	10	rabex 5333	. . . . 5 ⊢ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)} ∈ V
12	8, 9, 11	fvmpt 6999	. . . 4 ⊢ (𝑋 ∈ V → (Bnd‘𝑋) = {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)})
13	12	eleq2d 2817	. . 3 ⊢ (𝑋 ∈ V → (𝑀 ∈ (Bnd‘𝑋) ↔ 𝑀 ∈ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)}))
14		fveq2 6892	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (ball‘𝑚) = (ball‘𝑀))
15	14	oveqd 7430	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑥(ball‘𝑚)𝑟) = (𝑥(ball‘𝑀)𝑟))
16	15	eqeq2d 2741	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑋 = (𝑥(ball‘𝑚)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
17	16	rexbidv 3176	. . . . 5 ⊢ (𝑚 = 𝑀 → (∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟) ↔ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
18	17	ralbidv 3175	. . . 4 ⊢ (𝑚 = 𝑀 → (∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟) ↔ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
19	18	elrab 3684	. . 3 ⊢ (𝑀 ∈ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)} ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
20	13, 19	bitrdi 286	. 2 ⊢ (𝑋 ∈ V → (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))))
21	1, 3, 20	pm5.21nii 377	1 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))

Syntax hints:   ↔ wb 205   ∧ wa 394   = wceq 1539   ∈ wcel 2104  ∀wral 3059  ∃wrex 3068  {crab 3430  Vcvv 3472  ‘cfv 6544  (class class class)co 7413  ℝ⁺crp 12980  Metcmet 21132  ballcbl 21133  Bndcbnd 36940

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-iota 6496  df-fun 6546  df-fv 6552  df-ov 7416  df-bnd 36952

This theorem is referenced by:  bndmet  36954  isbndx  36955  isbnd3  36957  bndss  36959  totbndbnd  36962