Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  istotbnd3 Structured version   Visualization version   GIF version

Theorem istotbnd3 33541
Description: A metric space is totally bounded iff there is a finite ε-net for every positive ε. This differs from the definition in providing a finite set of ball centers rather than a finite set of balls. (Contributed by Mario Carneiro, 12-Sep-2015.)
Assertion
Ref Expression
istotbnd3 (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
Distinct variable groups:   𝑣,𝑑,𝑥,𝑀   𝑋,𝑑,𝑣,𝑥

Proof of Theorem istotbnd3
Dummy variables 𝑏 𝑓 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 istotbnd 33539 . 2 (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑤 ∈ Fin ( 𝑤 = 𝑋 ∧ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
2 oveq1 6642 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑏) → (𝑥(ball‘𝑀)𝑑) = ((𝑓𝑏)(ball‘𝑀)𝑑))
32eqeq2d 2630 . . . . . . . . . . 11 (𝑥 = (𝑓𝑏) → (𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
43ac6sfi 8189 . . . . . . . . . 10 ((𝑤 ∈ Fin ∧ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑓(𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
54ex 450 . . . . . . . . 9 (𝑤 ∈ Fin → (∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑓(𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))))
65ad2antlr 762 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ 𝑤 = 𝑋) → (∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑓(𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))))
7 simprrl 803 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → 𝑓:𝑤𝑋)
8 frn 6040 . . . . . . . . . . . . 13 (𝑓:𝑤𝑋 → ran 𝑓𝑋)
97, 8syl 17 . . . . . . . . . . . 12 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → ran 𝑓𝑋)
10 simplr 791 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → 𝑤 ∈ Fin)
11 ffn 6032 . . . . . . . . . . . . . . 15 (𝑓:𝑤𝑋𝑓 Fn 𝑤)
127, 11syl 17 . . . . . . . . . . . . . 14 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → 𝑓 Fn 𝑤)
13 dffn4 6108 . . . . . . . . . . . . . 14 (𝑓 Fn 𝑤𝑓:𝑤onto→ran 𝑓)
1412, 13sylib 208 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → 𝑓:𝑤onto→ran 𝑓)
15 fofi 8237 . . . . . . . . . . . . 13 ((𝑤 ∈ Fin ∧ 𝑓:𝑤onto→ran 𝑓) → ran 𝑓 ∈ Fin)
1610, 14, 15syl2anc 692 . . . . . . . . . . . 12 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → ran 𝑓 ∈ Fin)
17 elfpw 8253 . . . . . . . . . . . 12 (ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ↔ (ran 𝑓𝑋 ∧ ran 𝑓 ∈ Fin))
189, 16, 17sylanbrc 697 . . . . . . . . . . 11 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin))
192eleq2d 2685 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑓𝑏) → (𝑣 ∈ (𝑥(ball‘𝑀)𝑑) ↔ 𝑣 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑)))
2019rexrn 6347 . . . . . . . . . . . . . . 15 (𝑓 Fn 𝑤 → (∃𝑥 ∈ ran 𝑓 𝑣 ∈ (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑏𝑤 𝑣 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑)))
2112, 20syl 17 . . . . . . . . . . . . . 14 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → (∃𝑥 ∈ ran 𝑓 𝑣 ∈ (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑏𝑤 𝑣 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑)))
22 eliun 4515 . . . . . . . . . . . . . 14 (𝑣 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ ran 𝑓 𝑣 ∈ (𝑥(ball‘𝑀)𝑑))
23 eliun 4515 . . . . . . . . . . . . . 14 (𝑣 𝑏𝑤 ((𝑓𝑏)(ball‘𝑀)𝑑) ↔ ∃𝑏𝑤 𝑣 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑))
2421, 22, 233bitr4g 303 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → (𝑣 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ 𝑣 𝑏𝑤 ((𝑓𝑏)(ball‘𝑀)𝑑)))
2524eqrdv 2618 . . . . . . . . . . . 12 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = 𝑏𝑤 ((𝑓𝑏)(ball‘𝑀)𝑑))
26 simprrr 804 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))
27 iuneq2 4528 . . . . . . . . . . . . 13 (∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑) → 𝑏𝑤 𝑏 = 𝑏𝑤 ((𝑓𝑏)(ball‘𝑀)𝑑))
2826, 27syl 17 . . . . . . . . . . . 12 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → 𝑏𝑤 𝑏 = 𝑏𝑤 ((𝑓𝑏)(ball‘𝑀)𝑑))
29 uniiun 4564 . . . . . . . . . . . . 13 𝑤 = 𝑏𝑤 𝑏
30 simprl 793 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → 𝑤 = 𝑋)
3129, 30syl5eqr 2668 . . . . . . . . . . . 12 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → 𝑏𝑤 𝑏 = 𝑋)
3225, 28, 313eqtr2d 2660 . . . . . . . . . . 11 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = 𝑋)
33 iuneq1 4525 . . . . . . . . . . . . 13 (𝑣 = ran 𝑓 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑))
3433eqeq1d 2622 . . . . . . . . . . . 12 (𝑣 = ran 𝑓 → ( 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = 𝑋))
3534rspcev 3304 . . . . . . . . . . 11 ((ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = 𝑋) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)
3618, 32, 35syl2anc 692 . . . . . . . . . 10 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)
3736expr 642 . . . . . . . . 9 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ 𝑤 = 𝑋) → ((𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
3837exlimdv 1859 . . . . . . . 8 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ 𝑤 = 𝑋) → (∃𝑓(𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
396, 38syld 47 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ 𝑤 = 𝑋) → (∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
4039expimpd 628 . . . . . 6 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) → (( 𝑤 = 𝑋 ∧ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
4140rexlimdva 3027 . . . . 5 (𝑀 ∈ (Met‘𝑋) → (∃𝑤 ∈ Fin ( 𝑤 = 𝑋 ∧ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
42 elfpw 8253 . . . . . . . . . . 11 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣𝑋𝑣 ∈ Fin))
4342simprbi 480 . . . . . . . . . 10 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ∈ Fin)
4443ad2antrl 763 . . . . . . . . 9 ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → 𝑣 ∈ Fin)
45 mptfi 8250 . . . . . . . . 9 (𝑣 ∈ Fin → (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
46 rnfi 8234 . . . . . . . . 9 ((𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin → ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
4744, 45, 463syl 18 . . . . . . . 8 ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
48 ovex 6663 . . . . . . . . . 10 (𝑥(ball‘𝑀)𝑑) ∈ V
4948dfiun3 5369 . . . . . . . . 9 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑))
50 simprr 795 . . . . . . . . 9 ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)
5149, 50syl5eqr 2668 . . . . . . . 8 ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋)
52 eqid 2620 . . . . . . . . . 10 (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑))
5352rnmpt 5360 . . . . . . . . 9 ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = {𝑏 ∣ ∃𝑥𝑣 𝑏 = (𝑥(ball‘𝑀)𝑑)}
5442simplbi 476 . . . . . . . . . . . 12 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣𝑋)
5554ad2antrl 763 . . . . . . . . . . 11 ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → 𝑣𝑋)
56 ssrexv 3659 . . . . . . . . . . 11 (𝑣𝑋 → (∃𝑥𝑣 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
5755, 56syl 17 . . . . . . . . . 10 ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → (∃𝑥𝑣 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
5857ss2abdv 3667 . . . . . . . . 9 ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → {𝑏 ∣ ∃𝑥𝑣 𝑏 = (𝑥(ball‘𝑀)𝑑)} ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})
5953, 58syl5eqss 3641 . . . . . . . 8 ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})
60 unieq 4435 . . . . . . . . . . 11 (𝑤 = ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → 𝑤 = ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)))
6160eqeq1d 2622 . . . . . . . . . 10 (𝑤 = ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → ( 𝑤 = 𝑋 ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋))
62 ssabral 3665 . . . . . . . . . . 11 (𝑤 ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))
63 sseq1 3618 . . . . . . . . . . 11 (𝑤 = ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → (𝑤 ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)}))
6462, 63syl5bbr 274 . . . . . . . . . 10 (𝑤 = ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → (∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)}))
6561, 64anbi12d 746 . . . . . . . . 9 (𝑤 = ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → (( 𝑤 = 𝑋 ∧ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) ↔ ( ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋 ∧ ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})))
6665rspcev 3304 . . . . . . . 8 ((ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin ∧ ( ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋 ∧ ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)})) → ∃𝑤 ∈ Fin ( 𝑤 = 𝑋 ∧ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
6747, 51, 59, 66syl12anc 1322 . . . . . . 7 ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ∃𝑤 ∈ Fin ( 𝑤 = 𝑋 ∧ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)))
6867expr 642 . . . . . 6 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → ( 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 → ∃𝑤 ∈ Fin ( 𝑤 = 𝑋 ∧ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
6968rexlimdva 3027 . . . . 5 (𝑀 ∈ (Met‘𝑋) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 → ∃𝑤 ∈ Fin ( 𝑤 = 𝑋 ∧ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
7041, 69impbid 202 . . . 4 (𝑀 ∈ (Met‘𝑋) → (∃𝑤 ∈ Fin ( 𝑤 = 𝑋 ∧ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) ↔ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
7170ralbidv 2983 . . 3 (𝑀 ∈ (Met‘𝑋) → (∀𝑑 ∈ ℝ+𝑤 ∈ Fin ( 𝑤 = 𝑋 ∧ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
7271pm5.32i 668 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑤 ∈ Fin ( 𝑤 = 𝑋 ∧ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
731, 72bitri 264 1 (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wex 1702  wcel 1988  {cab 2606  wral 2909  wrex 2910  cin 3566  wss 3567  𝒫 cpw 4149   cuni 4427   ciun 4511  cmpt 4720  ran crn 5105   Fn wfn 5871  wf 5872  ontowfo 5874  cfv 5876  (class class class)co 6635  Fincfn 7940  +crp 11817  Metcme 19713  ballcbl 19714  TotBndctotbnd 33536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-int 4467  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-1o 7545  df-oadd 7549  df-er 7727  df-en 7941  df-dom 7942  df-fin 7944  df-totbnd 33538
This theorem is referenced by:  0totbnd  33543  sstotbnd2  33544  equivtotbnd  33548  totbndbnd  33559  prdstotbnd  33564
  Copyright terms: Public domain W3C validator