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Theorem istotbnd3 33700
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 33698 . 2 (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑤 ∈ Fin ( 𝑤 = 𝑋 ∧ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
2 oveq1 6697 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑏) → (𝑥(ball‘𝑀)𝑑) = ((𝑓𝑏)(ball‘𝑀)𝑑))
32eqeq2d 2661 . . . . . . . . . . 11 (𝑥 = (𝑓𝑏) → (𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
43ac6sfi 8245 . . . . . . . . . 10 ((𝑤 ∈ Fin ∧ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑓(𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
54ex 449 . . . . . . . . 9 (𝑤 ∈ Fin → (∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑓(𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))))
65ad2antlr 763 . . . . . . . 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 821 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → 𝑓:𝑤𝑋)
8 frn 6091 . . . . . . . . . . . . 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 807 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → 𝑤 ∈ Fin)
11 ffn 6083 . . . . . . . . . . . . . . 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 6159 . . . . . . . . . . . . . 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 8293 . . . . . . . . . . . . 13 ((𝑤 ∈ Fin ∧ 𝑓:𝑤onto→ran 𝑓) → ran 𝑓 ∈ Fin)
1610, 14, 15syl2anc 694 . . . . . . . . . . . 12 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → ran 𝑓 ∈ Fin)
17 elfpw 8309 . . . . . . . . . . . 12 (ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ↔ (ran 𝑓𝑋 ∧ ran 𝑓 ∈ Fin))
189, 16, 17sylanbrc 699 . . . . . . . . . . 11 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin))
192eleq2d 2716 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑓𝑏) → (𝑣 ∈ (𝑥(ball‘𝑀)𝑑) ↔ 𝑣 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑)))
2019rexrn 6401 . . . . . . . . . . . . . . 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 4556 . . . . . . . . . . . . . 14 (𝑣 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ ran 𝑓 𝑣 ∈ (𝑥(ball‘𝑀)𝑑))
23 eliun 4556 . . . . . . . . . . . . . 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 2649 . . . . . . . . . . . 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 822 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))
27 iuneq2 4569 . . . . . . . . . . . . 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 4605 . . . . . . . . . . . . 13 𝑤 = 𝑏𝑤 𝑏
30 simprl 809 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → 𝑤 = 𝑋)
3129, 30syl5eqr 2699 . . . . . . . . . . . 12 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → 𝑏𝑤 𝑏 = 𝑋)
3225, 28, 313eqtr2d 2691 . . . . . . . . . . 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 4566 . . . . . . . . . . . . 13 (𝑣 = ran 𝑓 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑))
3433eqeq1d 2653 . . . . . . . . . . . 12 (𝑣 = ran 𝑓 → ( 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = 𝑋))
3534rspcev 3340 . . . . . . . . . . 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 694 . . . . . . . . . 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 1901 . . . . . . . 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 3060 . . . . 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 8309 . . . . . . . . . . 11 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣𝑋𝑣 ∈ Fin))
4342simprbi 479 . . . . . . . . . 10 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ∈ Fin)
4443ad2antrl 764 . . . . . . . . 9 ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → 𝑣 ∈ Fin)
45 mptfi 8306 . . . . . . . . 9 (𝑣 ∈ Fin → (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
46 rnfi 8290 . . . . . . . . 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 6718 . . . . . . . . . 10 (𝑥(ball‘𝑀)𝑑) ∈ V
4948dfiun3 5412 . . . . . . . . 9 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑))
50 simprr 811 . . . . . . . . 9 ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)
5149, 50syl5eqr 2699 . . . . . . . 8 ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋)
52 eqid 2651 . . . . . . . . . 10 (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑))
5352rnmpt 5403 . . . . . . . . 9 ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = {𝑏 ∣ ∃𝑥𝑣 𝑏 = (𝑥(ball‘𝑀)𝑑)}
5442simplbi 475 . . . . . . . . . . . 12 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣𝑋)
5554ad2antrl 764 . . . . . . . . . . 11 ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → 𝑣𝑋)
56 ssrexv 3700 . . . . . . . . . . 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 3708 . . . . . . . . 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 3682 . . . . . . . 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 4476 . . . . . . . . . . 11 (𝑤 = ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → 𝑤 = ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)))
6160eqeq1d 2653 . . . . . . . . . 10 (𝑤 = ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → ( 𝑤 = 𝑋 ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋))
62 ssabral 3706 . . . . . . . . . . 11 (𝑤 ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))
63 sseq1 3659 . . . . . . . . . . 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 747 . . . . . . . . 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 3340 . . . . . . . 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 1364 . . . . . . 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 3060 . . . . 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 3015 . . 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 670 . 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 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wral 2941  wrex 2942  cin 3606  wss 3607  𝒫 cpw 4191   cuni 4468   ciun 4552  cmpt 4762  ran crn 5144   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  (class class class)co 6690  Fincfn 7997  +crp 11870  Metcme 19780  ballcbl 19781  TotBndctotbnd 33695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-en 7998  df-dom 7999  df-fin 8001  df-totbnd 33697
This theorem is referenced by:  0totbnd  33702  sstotbnd2  33703  equivtotbnd  33707  totbndbnd  33718  prdstotbnd  33723
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