Theorem 0totbnd 35858

Description: The metric (there is only one) on the empty set is totally bounded. (Contributed by Mario Carneiro, 16-Sep-2015.)

Assertion

Ref	Expression
0totbnd	⊢ (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (Met‘𝑋)))

Proof of Theorem 0totbnd

Dummy variables 𝑣 𝑟 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6756	. . 3 ⊢ (𝑋 = ∅ → (TotBnd‘𝑋) = (TotBnd‘∅))
2	1	eleq2d 2824	. 2 ⊢ (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (TotBnd‘∅)))
3		0elpw 5273	. . . . . . 7 ⊢ ∅ ∈ 𝒫 ∅
4		0fin 8916	. . . . . . 7 ⊢ ∅ ∈ Fin
5		elin 3899	. . . . . . 7 ⊢ (∅ ∈ (𝒫 ∅ ∩ Fin) ↔ (∅ ∈ 𝒫 ∅ ∧ ∅ ∈ Fin))
6	3, 4, 5	mpbir2an 707	. . . . . 6 ⊢ ∅ ∈ (𝒫 ∅ ∩ Fin)
7		0iun 4988	. . . . . 6 ⊢ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅
8		iuneq1 4937	. . . . . . . 8 ⊢ (𝑣 = ∅ → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟))
9	8	eqeq1d 2740	. . . . . . 7 ⊢ (𝑣 = ∅ → (∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅ ↔ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅))
10	9	rspcev 3552	. . . . . 6 ⊢ ((∅ ∈ (𝒫 ∅ ∩ Fin) ∧ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅) → ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅)
11	6, 7, 10	mp2an 688	. . . . 5 ⊢ ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
12	11	rgenw 3075	. . . 4 ⊢ ∀𝑟 ∈ ℝ+ ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
13		istotbnd3 35856	. . . 4 ⊢ (𝑀 ∈ (TotBnd‘∅) ↔ (𝑀 ∈ (Met‘∅) ∧ ∀𝑟 ∈ ℝ+ ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅))
14	12, 13	mpbiran2 706	. . 3 ⊢ (𝑀 ∈ (TotBnd‘∅) ↔ 𝑀 ∈ (Met‘∅))
15		fveq2 6756	. . . 4 ⊢ (𝑋 = ∅ → (Met‘𝑋) = (Met‘∅))
16	15	eleq2d 2824	. . 3 ⊢ (𝑋 = ∅ → (𝑀 ∈ (Met‘𝑋) ↔ 𝑀 ∈ (Met‘∅)))
17	14, 16	bitr4id 289	. 2 ⊢ (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘∅) ↔ 𝑀 ∈ (Met‘𝑋)))
18	2, 17	bitrd 278	1 ⊢ (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (Met‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ∩ cin 3882  ∅c0 4253  𝒫 cpw 4530  ∪ ciun 4921  ‘cfv 6418  (class class class)co 7255  Fincfn 8691  ℝ⁺crp 12659  Metcmet 20496  ballcbl 20497  TotBndctotbnd 35851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-om 7688  df-1st 7804  df-2nd 7805  df-1o 8267  df-er 8456  df-en 8692  df-dom 8693  df-fin 8695  df-totbnd 35853

This theorem is referenced by:  prdsbnd2  35880