Theorem 0totbnd 38523

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 6878	. . 3 ⊢ (𝑋 = ∅ → (TotBnd‘𝑋) = (TotBnd‘∅))
2	1	eleq2d 2846	. 2 ⊢ (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (TotBnd‘∅)))
3		0elpw 5320	. . . . . . 7 ⊢ ∅ ∈ 𝒫 ∅
4		0fi 9049	. . . . . . 7 ⊢ ∅ ∈ Fin
5		elin 3915	. . . . . . 7 ⊢ (∅ ∈ (𝒫 ∅ ∩ Fin) ↔ (∅ ∈ 𝒫 ∅ ∧ ∅ ∈ Fin))
6	3, 4, 5	mpbir2an 724	. . . . . 6 ⊢ ∅ ∈ (𝒫 ∅ ∩ Fin)
7		0iun 5021	. . . . . 6 ⊢ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅
8		iuneq1 4968	. . . . . . . 8 ⊢ (𝑣 = ∅ → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟))
9	8	eqeq1d 2762	. . . . . . 7 ⊢ (𝑣 = ∅ → (∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅ ↔ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅))
10	9	rspcev 3576	. . . . . 6 ⊢ ((∅ ∈ (𝒫 ∅ ∩ Fin) ∧ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅) → ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅)
11	6, 7, 10	mp2an 705	. . . . 5 ⊢ ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
12	11	rgenw 3080	. . . 4 ⊢ ∀𝑟 ∈ ℝ+ ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
13		istotbnd3 38521	. . . 4 ⊢ (𝑀 ∈ (TotBnd‘∅) ↔ (𝑀 ∈ (Met‘∅) ∧ ∀𝑟 ∈ ℝ+ ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅))
14	12, 13	mpbiran2 723	. . 3 ⊢ (𝑀 ∈ (TotBnd‘∅) ↔ 𝑀 ∈ (Met‘∅))
15		fveq2 6878	. . . 4 ⊢ (𝑋 = ∅ → (Met‘𝑋) = (Met‘∅))
16	15	eleq2d 2846	. . 3 ⊢ (𝑋 = ∅ → (𝑀 ∈ (Met‘𝑋) ↔ 𝑀 ∈ (Met‘∅)))
17	14, 16	bitr4id 293	. 2 ⊢ (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘∅) ↔ 𝑀 ∈ (Met‘𝑋)))
18	2, 17	bitrd 282	1 ⊢ (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (Met‘𝑋)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ∩ cin 3898  ∅c0 4279  𝒫 cpw 4557  ∪ ciun 4951  ‘cfv 6533  (class class class)co 7413  Fincfn 8952  ℝ⁺crp 13042  Metcmet 21571  ballcbl 21572  TotBndctotbnd 38516

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7736

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7416  df-om 7863  df-1st 7986  df-2nd 7987  df-1o 8455  df-en 8953  df-dom 8954  df-fin 8956  df-totbnd 38518

This theorem is used by:  prdsbnd2  38545