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Theorem istotbnd3 33902
 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 33900 . 2 (𝑀 ∈ (TotBnd‘𝑋) ↔ (𝑀 ∈ (Met‘𝑋) ∧ ∀𝑑 ∈ ℝ+𝑤 ∈ Fin ( 𝑤 = 𝑋 ∧ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))))
2 oveq1 6800 . . . . . . . . . . . 12 (𝑥 = (𝑓𝑏) → (𝑥(ball‘𝑀)𝑑) = ((𝑓𝑏)(ball‘𝑀)𝑑))
32eqeq2d 2781 . . . . . . . . . . 11 (𝑥 = (𝑓𝑏) → (𝑏 = (𝑥(ball‘𝑀)𝑑) ↔ 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
43ac6sfi 8360 . . . . . . . . . 10 ((𝑤 ∈ Fin ∧ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)) → ∃𝑓(𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))
54ex 397 . . . . . . . . 9 (𝑤 ∈ Fin → (∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑) → ∃𝑓(𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))))
65ad2antlr 706 . . . . . . . 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 766 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → 𝑓:𝑤𝑋)
8 frn 6193 . . . . . . . . . . . . 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 752 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → 𝑤 ∈ Fin)
11 ffn 6185 . . . . . . . . . . . . . . 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 6262 . . . . . . . . . . . . . 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 8408 . . . . . . . . . . . . 13 ((𝑤 ∈ Fin ∧ 𝑓:𝑤onto→ran 𝑓) → ran 𝑓 ∈ Fin)
1610, 14, 15syl2anc 573 . . . . . . . . . . . 12 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → ran 𝑓 ∈ Fin)
17 elfpw 8424 . . . . . . . . . . . 12 (ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin) ↔ (ran 𝑓𝑋 ∧ ran 𝑓 ∈ Fin))
189, 16, 17sylanbrc 572 . . . . . . . . . . 11 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → ran 𝑓 ∈ (𝒫 𝑋 ∩ Fin))
192eleq2d 2836 . . . . . . . . . . . . . . . 16 (𝑥 = (𝑓𝑏) → (𝑣 ∈ (𝑥(ball‘𝑀)𝑑) ↔ 𝑣 ∈ ((𝑓𝑏)(ball‘𝑀)𝑑)))
2019rexrn 6504 . . . . . . . . . . . . . . 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 4658 . . . . . . . . . . . . . 14 (𝑣 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥 ∈ ran 𝑓 𝑣 ∈ (𝑥(ball‘𝑀)𝑑))
23 eliun 4658 . . . . . . . . . . . . . 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 2769 . . . . . . . . . . . 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 767 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑))
27 iuneq2 4671 . . . . . . . . . . . . 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 4707 . . . . . . . . . . . . 13 𝑤 = 𝑏𝑤 𝑏
30 simprl 754 . . . . . . . . . . . . 13 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → 𝑤 = 𝑋)
3129, 30syl5eqr 2819 . . . . . . . . . . . 12 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑤 ∈ Fin) ∧ ( 𝑤 = 𝑋 ∧ (𝑓:𝑤𝑋 ∧ ∀𝑏𝑤 𝑏 = ((𝑓𝑏)(ball‘𝑀)𝑑)))) → 𝑏𝑤 𝑏 = 𝑋)
3225, 28, 313eqtr2d 2811 . . . . . . . . . . 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 4668 . . . . . . . . . . . . 13 (𝑣 = ran 𝑓 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑))
3433eqeq1d 2773 . . . . . . . . . . . 12 (𝑣 = ran 𝑓 → ( 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑀)𝑑) = 𝑋))
3534rspcev 3460 . . . . . . . . . . 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 573 . . . . . . . . . 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 444 . . . . . . . . 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 2013 . . . . . . . 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 441 . . . . . 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 3179 . . . . 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 8424 . . . . . . . . . . 11 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣𝑋𝑣 ∈ Fin))
4342simprbi 484 . . . . . . . . . 10 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ∈ Fin)
4443ad2antrl 707 . . . . . . . . 9 ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → 𝑣 ∈ Fin)
45 mptfi 8421 . . . . . . . . 9 (𝑣 ∈ Fin → (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) ∈ Fin)
46 rnfi 8405 . . . . . . . . 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 6823 . . . . . . . . . 10 (𝑥(ball‘𝑀)𝑑) ∈ V
4948dfiun3 5518 . . . . . . . . 9 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑))
50 simprr 756 . . . . . . . . 9 ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)
5149, 50syl5eqr 2819 . . . . . . . 8 ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋)
52 eqid 2771 . . . . . . . . . 10 (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑))
5352rnmpt 5509 . . . . . . . . 9 ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = {𝑏 ∣ ∃𝑥𝑣 𝑏 = (𝑥(ball‘𝑀)𝑑)}
5442simplbi 485 . . . . . . . . . . . 12 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣𝑋)
5554ad2antrl 707 . . . . . . . . . . 11 ((𝑀 ∈ (Met‘𝑋) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑋)) → 𝑣𝑋)
56 ssrexv 3816 . . . . . . . . . . 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 3824 . . . . . . . . 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 3798 . . . . . . . 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 4582 . . . . . . . . . . 11 (𝑤 = ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → 𝑤 = ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)))
6160eqeq1d 2773 . . . . . . . . . 10 (𝑤 = ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) → ( 𝑤 = 𝑋 ran (𝑥𝑣 ↦ (𝑥(ball‘𝑀)𝑑)) = 𝑋))
62 ssabral 3822 . . . . . . . . . . 11 (𝑤 ⊆ {𝑏 ∣ ∃𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑)} ↔ ∀𝑏𝑤𝑥𝑋 𝑏 = (𝑥(ball‘𝑀)𝑑))
63 sseq1 3775 . . . . . . . . . . 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 616 . . . . . . . . 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 3460 . . . . . . . 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 1474 . . . . . . 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 444 . . . . . 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 3179 . . . . 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 3135 . . 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 564 . 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 382   = wceq 1631  ∃wex 1852   ∈ wcel 2145  {cab 2757  ∀wral 3061  ∃wrex 3062   ∩ cin 3722   ⊆ wss 3723  𝒫 cpw 4297  ∪ cuni 4574  ∪ ciun 4654   ↦ cmpt 4863  ran crn 5250   Fn wfn 6026  ⟶wf 6027  –onto→wfo 6029  ‘cfv 6031  (class class class)co 6793  Fincfn 8109  ℝ+crp 12035  Metcme 19947  ballcbl 19948  TotBndctotbnd 33897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-1o 7713  df-oadd 7717  df-er 7896  df-en 8110  df-dom 8111  df-fin 8113  df-totbnd 33899 This theorem is referenced by:  0totbnd  33904  sstotbnd2  33905  equivtotbnd  33909  totbndbnd  33920  prdstotbnd  33925
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