Theorem 0totbnd 38108

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 6834	. . 3 ⊢ (𝑋 = ∅ → (TotBnd‘𝑋) = (TotBnd‘∅))
2	1	eleq2d 2823	. 2 ⊢ (𝑋 = ∅ → (𝑀 ∈ (TotBnd‘𝑋) ↔ 𝑀 ∈ (TotBnd‘∅)))
3		0elpw 5293	. . . . . . 7 ⊢ ∅ ∈ 𝒫 ∅
4		0fi 8982	. . . . . . 7 ⊢ ∅ ∈ Fin
5		elin 3906	. . . . . . 7 ⊢ (∅ ∈ (𝒫 ∅ ∩ Fin) ↔ (∅ ∈ 𝒫 ∅ ∧ ∅ ∈ Fin))
6	3, 4, 5	mpbir2an 712	. . . . . 6 ⊢ ∅ ∈ (𝒫 ∅ ∩ Fin)
7		0iun 5006	. . . . . 6 ⊢ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅
8		iuneq1 4951	. . . . . . . 8 ⊢ (𝑣 = ∅ → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟))
9	8	eqeq1d 2739	. . . . . . 7 ⊢ (𝑣 = ∅ → (∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅ ↔ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅))
10	9	rspcev 3565	. . . . . 6 ⊢ ((∅ ∈ (𝒫 ∅ ∩ Fin) ∧ ∪ 𝑥 ∈ ∅ (𝑥(ball‘𝑀)𝑟) = ∅) → ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅)
11	6, 7, 10	mp2an 693	. . . . 5 ⊢ ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
12	11	rgenw 3056	. . . 4 ⊢ ∀𝑟 ∈ ℝ+ ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅
13		istotbnd3 38106	. . . 4 ⊢ (𝑀 ∈ (TotBnd‘∅) ↔ (𝑀 ∈ (Met‘∅) ∧ ∀𝑟 ∈ ℝ+ ∃𝑣 ∈ (𝒫 ∅ ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑟) = ∅))
14	12, 13	mpbiran2 711	. . 3 ⊢ (𝑀 ∈ (TotBnd‘∅) ↔ 𝑀 ∈ (Met‘∅))
15		fveq2 6834	. . . 4 ⊢ (𝑋 = ∅ → (Met‘𝑋) = (Met‘∅))
16	15	eleq2d 2823	. . 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 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062   ∩ cin 3889  ∅c0 4274  𝒫 cpw 4542  ∪ ciun 4934  ‘cfv 6492  (class class class)co 7360  Fincfn 8886  ℝ⁺crp 12933  Metcmet 21330  ballcbl 21331  TotBndctotbnd 38101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-om 7811  df-1st 7935  df-2nd 7936  df-1o 8398  df-en 8887  df-dom 8888  df-fin 8890  df-totbnd 38103

This theorem is referenced by:  prdsbnd2  38130