Theorem isbnd 37843

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 6865	. 2 ⊢ (𝑀 ∈ (Bnd‘𝑋) → 𝑋 ∈ V)
2		elfvex 6865	. . 3 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑋 ∈ V)
3	2	adantr 480	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)) → 𝑋 ∈ V)
4		fveq2 6830	. . . . . 6 ⊢ (𝑦 = 𝑋 → (Met‘𝑦) = (Met‘𝑋))
5		eqeq1 2737	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑦 = (𝑥(ball‘𝑚)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑚)𝑟)))
6	5	rexbidv 3157	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (∃𝑟 ∈ ℝ+ 𝑦 = (𝑥(ball‘𝑚)𝑟) ↔ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)))
7	6	raleqbi1dv 3305	. . . . . 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 3414	. . . . 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 37842	. . . . 5 ⊢ Bnd = (𝑦 ∈ V ↦ {𝑚 ∈ (Met‘𝑦) ∣ ∀𝑥 ∈ 𝑦 ∃𝑟 ∈ ℝ+ 𝑦 = (𝑥(ball‘𝑚)𝑟)})
10		fvex 6843	. . . . . 6 ⊢ (Met‘𝑋) ∈ V
11	10	rabex 5281	. . . . 5 ⊢ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)} ∈ V
12	8, 9, 11	fvmpt 6937	. . . 4 ⊢ (𝑋 ∈ V → (Bnd‘𝑋) = {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)})
13	12	eleq2d 2819	. . 3 ⊢ (𝑋 ∈ V → (𝑀 ∈ (Bnd‘𝑋) ↔ 𝑀 ∈ {𝑚 ∈ (Met‘𝑋) ∣ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟)}))
14		fveq2 6830	. . . . . . . 8 ⊢ (𝑚 = 𝑀 → (ball‘𝑚) = (ball‘𝑀))
15	14	oveqd 7371	. . . . . . 7 ⊢ (𝑚 = 𝑀 → (𝑥(ball‘𝑚)𝑟) = (𝑥(ball‘𝑀)𝑟))
16	15	eqeq2d 2744	. . . . . 6 ⊢ (𝑚 = 𝑀 → (𝑋 = (𝑥(ball‘𝑚)𝑟) ↔ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
17	16	rexbidv 3157	. . . . 5 ⊢ (𝑚 = 𝑀 → (∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟) ↔ ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
18	17	ralbidv 3156	. . . 4 ⊢ (𝑚 = 𝑀 → (∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑚)𝑟) ↔ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))
19	18	elrab 3643	. . 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 287	. 2 ⊢ (𝑋 ∈ V → (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟))))
21	1, 3, 20	pm5.21nii 378	1 ⊢ (𝑀 ∈ (Bnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑥 ∈ 𝑋 ∃𝑟 ∈ ℝ+ 𝑋 = (𝑥(ball‘𝑀)𝑟)))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∀wral 3048  ∃wrex 3057  {crab 3396  Vcvv 3437  ‘cfv 6488  (class class class)co 7354  ℝ⁺crp 12894  Metcmet 21281  ballcbl 21282  Bndcbnd 37830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6444  df-fun 6490  df-fv 6496  df-ov 7357  df-bnd 37842

This theorem is referenced by:  bndmet  37844  isbndx  37845  isbnd3  37847  bndss  37849  totbndbnd  37852