Theorem 0totbnd 37772

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 6826	. . 3 ⊢ (𝑋 = ∅ → (TotBnd‘𝑋) = (TotBnd‘∅))
2	1	eleq2d 2814	. 2 ⊢ (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (TotBnd‘∅)))
3		0elpw 5298	. . . . . . 7 ⊢ ∅ ∈ 𝒫 ∅
4		0fi 8974	. . . . . . 7 ⊢ ∅ ∈ Fin
5		elin 3921	. . . . . . 7 ⊢ (∅ ∈ (𝒫 ∅ ∩ Fin) ↔ (∅ ∈ 𝒫 ∅ ∧ ∅ ∈ Fin))
6	3, 4, 5	mpbir2an 711	. . . . . 6 ⊢ ∅ ∈ (𝒫 ∅ ∩ Fin)
7		0iun 5015	. . . . . 6 ⊢ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅
8		iuneq1 4961	. . . . . . . 8 ⊢ (𝑣 = ∅ → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟))
9	8	eqeq1d 2731	. . . . . . 7 ⊢ (𝑣 = ∅ → (∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅ ↔ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅))
10	9	rspcev 3579	. . . . . 6 ⊢ ((∅ ∈ (𝒫 ∅ ∩ Fin) ∧ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅) → ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅)
11	6, 7, 10	mp2an 692	. . . . 5 ⊢ ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
12	11	rgenw 3048	. . . 4 ⊢ ∀𝑟 ∈ ℝ+ ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
13		istotbnd3 37770	. . . 4 ⊢ (𝑀 ∈ (TotBnd‘∅) ↔ (𝑀 ∈ (Met‘∅) ∧ ∀𝑟 ∈ ℝ+ ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅))
14	12, 13	mpbiran2 710	. . 3 ⊢ (𝑀 ∈ (TotBnd‘∅) ↔ 𝑀 ∈ (Met‘∅))
15		fveq2 6826	. . . 4 ⊢ (𝑋 = ∅ → (Met‘𝑋) = (Met‘∅))
16	15	eleq2d 2814	. . 3 ⊢ (𝑋 = ∅ → (𝑀 ∈ (Met‘𝑋) ↔ 𝑀 ∈ (Met‘∅)))
17	14, 16	bitr4id 290	. 2 ⊢ (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘∅) ↔ 𝑀 ∈ (Met‘𝑋)))
18	2, 17	bitrd 279	1 ⊢ (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (Met‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ∩ cin 3904  ∅c0 4286  𝒫 cpw 4553  ∪ ciun 4944  ‘cfv 6486  (class class class)co 7353  Fincfn 8879  ℝ⁺crp 12912  Metcmet 21266  ballcbl 21267  TotBndctotbnd 37765

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-om 7807  df-1st 7931  df-2nd 7932  df-1o 8395  df-en 8880  df-dom 8881  df-fin 8883  df-totbnd 37767

This theorem is referenced by:  prdsbnd2  37794