Theorem 0totbnd 33572

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 6191	. . 3 ⊢ (𝑋 = ∅ → (TotBnd‘𝑋) = (TotBnd‘∅))
2	1	eleq2d 2687	. 2 ⊢ (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (TotBnd‘∅)))
3		fveq2 6191	. . . 4 ⊢ (𝑋 = ∅ → (Met‘𝑋) = (Met‘∅))
4	3	eleq2d 2687	. . 3 ⊢ (𝑋 = ∅ → (𝑀 ∈ (Met‘𝑋) ↔ 𝑀 ∈ (Met‘∅)))
5		0elpw 4834	. . . . . . 7 ⊢ ∅ ∈ 𝒫 ∅
6		0fin 8188	. . . . . . 7 ⊢ ∅ ∈ Fin
7		elin 3796	. . . . . . 7 ⊢ (∅ ∈ (𝒫 ∅ ∩ Fin) ↔ (∅ ∈ 𝒫 ∅ ∧ ∅ ∈ Fin))
8	5, 6, 7	mpbir2an 955	. . . . . 6 ⊢ ∅ ∈ (𝒫 ∅ ∩ Fin)
9		0iun 4577	. . . . . 6 ⊢ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅
10		iuneq1 4534	. . . . . . . 8 ⊢ (𝑣 = ∅ → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟))
11	10	eqeq1d 2624	. . . . . . 7 ⊢ (𝑣 = ∅ → (∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅ ↔ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅))
12	11	rspcev 3309	. . . . . 6 ⊢ ((∅ ∈ (𝒫 ∅ ∩ Fin) ∧ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅) → ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅)
13	8, 9, 12	mp2an 708	. . . . 5 ⊢ ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
14	13	rgenw 2924	. . . 4 ⊢ ∀𝑟 ∈ ℝ+ ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
15		istotbnd3 33570	. . . 4 ⊢ (𝑀 ∈ (TotBnd‘∅) ↔ (𝑀 ∈ (Met‘∅) ∧ ∀𝑟 ∈ ℝ+ ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅))
16	14, 15	mpbiran2 954	. . 3 ⊢ (𝑀 ∈ (TotBnd‘∅) ↔ 𝑀 ∈ (Met‘∅))
17	4, 16	syl6rbbr 279	. 2 ⊢ (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘∅) ↔ 𝑀 ∈ (Met‘𝑋)))
18	2, 17	bitrd 268	1 ⊢ (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (Met‘𝑋)))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∩ cin 3573  ∅c0 3915  𝒫 cpw 4158  ∪ ciun 4520  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  ℝ⁺crp 11832  Metcme 19732  ballcbl 19733  TotBndctotbnd 33565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-fin 7959  df-totbnd 33567

This theorem is referenced by:  prdsbnd2  33594