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Theorem sstotbnd2 33240
Description: Condition for a subset of a metric space to be totally bounded. (Contributed by Mario Carneiro, 12-Sep-2015.)
Hypothesis
Ref Expression
sstotbnd.2 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))
Assertion
Ref Expression
sstotbnd2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑)))
Distinct variable groups:   𝑣,𝑑,𝑥,𝑀   𝑋,𝑑,𝑣,𝑥   𝑁,𝑑,𝑣,𝑥   𝑌,𝑑,𝑣,𝑥

Proof of Theorem sstotbnd2
Dummy variables 𝑐 𝑓 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sstotbnd.2 . . . . 5 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))
2 metres2 22091 . . . . 5 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑀 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌))
31, 2syl5eqel 2702 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → 𝑁 ∈ (Met‘𝑌))
4 istotbnd3 33237 . . . . 5 (𝑁 ∈ (TotBnd‘𝑌) ↔ (𝑁 ∈ (Met‘𝑌) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌))
54baib 943 . . . 4 (𝑁 ∈ (Met‘𝑌) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌))
63, 5syl 17 . . 3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌))
7 simpllr 798 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → 𝑌𝑋)
8 sspwb 4883 . . . . . . . . . 10 (𝑌𝑋 ↔ 𝒫 𝑌 ⊆ 𝒫 𝑋)
97, 8sylib 208 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → 𝒫 𝑌 ⊆ 𝒫 𝑋)
10 ssrin 3821 . . . . . . . . 9 (𝒫 𝑌 ⊆ 𝒫 𝑋 → (𝒫 𝑌 ∩ Fin) ⊆ (𝒫 𝑋 ∩ Fin))
119, 10syl 17 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → (𝒫 𝑌 ∩ Fin) ⊆ (𝒫 𝑋 ∩ Fin))
12 simprl 793 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → 𝑣 ∈ (𝒫 𝑌 ∩ Fin))
1311, 12sseldd 3588 . . . . . . 7 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → 𝑣 ∈ (𝒫 𝑋 ∩ Fin))
14 simprr 795 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)
15 metxmet 22062 . . . . . . . . . . . . . 14 (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
1615ad4antr 767 . . . . . . . . . . . . 13 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → 𝑀 ∈ (∞Met‘𝑋))
17 elfpw 8220 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ↔ (𝑣𝑌𝑣 ∈ Fin))
1817simplbi 476 . . . . . . . . . . . . . . . 16 (𝑣 ∈ (𝒫 𝑌 ∩ Fin) → 𝑣𝑌)
1918adantl 482 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) → 𝑣𝑌)
2019sselda 3587 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → 𝑥𝑌)
21 simp-4r 806 . . . . . . . . . . . . . . 15 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → 𝑌𝑋)
22 sseqin2 3800 . . . . . . . . . . . . . . 15 (𝑌𝑋 ↔ (𝑋𝑌) = 𝑌)
2321, 22sylib 208 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → (𝑋𝑌) = 𝑌)
2420, 23eleqtrrd 2701 . . . . . . . . . . . . 13 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → 𝑥 ∈ (𝑋𝑌))
25 simpllr 798 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → 𝑑 ∈ ℝ+)
2625rpxrd 11825 . . . . . . . . . . . . 13 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → 𝑑 ∈ ℝ*)
271blres 22159 . . . . . . . . . . . . 13 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ (𝑋𝑌) ∧ 𝑑 ∈ ℝ*) → (𝑥(ball‘𝑁)𝑑) = ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌))
2816, 24, 26, 27syl3anc 1323 . . . . . . . . . . . 12 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → (𝑥(ball‘𝑁)𝑑) = ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌))
29 inss1 3816 . . . . . . . . . . . 12 ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ⊆ (𝑥(ball‘𝑀)𝑑)
3028, 29syl6eqss 3639 . . . . . . . . . . 11 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → (𝑥(ball‘𝑁)𝑑) ⊆ (𝑥(ball‘𝑀)𝑑))
3130ralrimiva 2961 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) → ∀𝑥𝑣 (𝑥(ball‘𝑁)𝑑) ⊆ (𝑥(ball‘𝑀)𝑑))
32 ss2iun 4507 . . . . . . . . . 10 (∀𝑥𝑣 (𝑥(ball‘𝑁)𝑑) ⊆ (𝑥(ball‘𝑀)𝑑) → 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) ⊆ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑))
3331, 32syl 17 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) → 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) ⊆ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑))
3433adantrr 752 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) ⊆ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑))
3514, 34eqsstr3d 3624 . . . . . . 7 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑))
3613, 35jca 554 . . . . . 6 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑)))
3736ex 450 . . . . 5 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) → ((𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌) → (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑))))
3837reximdv2 3009 . . . 4 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) → (∃𝑣 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌 → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑)))
3938ralimdva 2957 . . 3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌 → ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑)))
406, 39sylbid 230 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) → ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑)))
41 simpr 477 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) → 𝑐 ∈ ℝ+)
4241rphalfcld 11836 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) → (𝑐 / 2) ∈ ℝ+)
43 oveq2 6618 . . . . . . . . . 10 (𝑑 = (𝑐 / 2) → (𝑥(ball‘𝑀)𝑑) = (𝑥(ball‘𝑀)(𝑐 / 2)))
4443iuneq2d 4518 . . . . . . . . 9 (𝑑 = (𝑐 / 2) → 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))
4544sseq2d 3617 . . . . . . . 8 (𝑑 = (𝑐 / 2) → (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ↔ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2))))
4645rexbidv 3046 . . . . . . 7 (𝑑 = (𝑐 / 2) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2))))
4746rspcv 3294 . . . . . 6 ((𝑐 / 2) ∈ ℝ+ → (∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2))))
4842, 47syl 17 . . . . 5 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) → (∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2))))
49 elfpw 8220 . . . . . . . . . . 11 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣𝑋𝑣 ∈ Fin))
5049simprbi 480 . . . . . . . . . 10 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ∈ Fin)
5150ad2antrl 763 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → 𝑣 ∈ Fin)
52 ssrab2 3671 . . . . . . . . 9 {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ⊆ 𝑣
53 ssfi 8132 . . . . . . . . 9 ((𝑣 ∈ Fin ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ⊆ 𝑣) → {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ∈ Fin)
5451, 52, 53sylancl 693 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ∈ Fin)
55 oveq1 6617 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (𝑥(ball‘𝑀)(𝑐 / 2)) = (𝑦(ball‘𝑀)(𝑐 / 2)))
5655ineq1d 3796 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌))
57 incom 3788 . . . . . . . . . . . . . . 15 ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = (𝑌 ∩ (𝑦(ball‘𝑀)(𝑐 / 2)))
5856, 57syl6eq 2671 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = (𝑌 ∩ (𝑦(ball‘𝑀)(𝑐 / 2))))
59 dfin5 3567 . . . . . . . . . . . . . 14 (𝑌 ∩ (𝑦(ball‘𝑀)(𝑐 / 2))) = {𝑧𝑌𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))}
6058, 59syl6eq 2671 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = {𝑧𝑌𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))})
6160neeq1d 2849 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ ↔ {𝑧𝑌𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))} ≠ ∅))
62 rabn0 3937 . . . . . . . . . . . 12 ({𝑧𝑌𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))} ≠ ∅ ↔ ∃𝑧𝑌 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))
6361, 62syl6bb 276 . . . . . . . . . . 11 (𝑥 = 𝑦 → (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ ↔ ∃𝑧𝑌 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
6463elrab 3350 . . . . . . . . . 10 (𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ↔ (𝑦𝑣 ∧ ∃𝑧𝑌 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
6564simprbi 480 . . . . . . . . 9 (𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} → ∃𝑧𝑌 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))
6665rgen 2917 . . . . . . . 8 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}∃𝑧𝑌 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))
67 eleq1 2686 . . . . . . . . 9 (𝑧 = (𝑓𝑦) → (𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2)) ↔ (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
6867ac6sfi 8156 . . . . . . . 8 (({𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ∈ Fin ∧ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}∃𝑧𝑌 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))) → ∃𝑓(𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
6954, 66, 68sylancl 693 . . . . . . 7 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → ∃𝑓(𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
70 fdm 6013 . . . . . . . . . . . . . 14 (𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 → dom 𝑓 = {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅})
7170ad2antrl 763 . . . . . . . . . . . . 13 ((𝑣 ∈ Fin ∧ (𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → dom 𝑓 = {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅})
7271, 52syl6eqss 3639 . . . . . . . . . . . 12 ((𝑣 ∈ Fin ∧ (𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → dom 𝑓𝑣)
73 simprl 793 . . . . . . . . . . . . 13 ((𝑣 ∈ Fin ∧ (𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → 𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌)
7471feq2d 5993 . . . . . . . . . . . . 13 ((𝑣 ∈ Fin ∧ (𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑓:dom 𝑓𝑌𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌))
7573, 74mpbird 247 . . . . . . . . . . . 12 ((𝑣 ∈ Fin ∧ (𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → 𝑓:dom 𝑓𝑌)
76 simprr 795 . . . . . . . . . . . . . 14 ((𝑣 ∈ Fin ∧ (𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))
77 ffn 6007 . . . . . . . . . . . . . . . . . 18 (𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌𝑓 Fn {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅})
78 elpreima 6298 . . . . . . . . . . . . . . . . . 18 (𝑓 Fn {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} → (𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ (𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ∧ (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))))
7977, 78syl 17 . . . . . . . . . . . . . . . . 17 (𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 → (𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ (𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ∧ (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))))
8079baibd 947 . . . . . . . . . . . . . . . 16 ((𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}) → (𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
8180ralbidva 2980 . . . . . . . . . . . . . . 15 (𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 → (∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
8281ad2antrl 763 . . . . . . . . . . . . . 14 ((𝑣 ∈ Fin ∧ (𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
8376, 82mpbird 247 . . . . . . . . . . . . 13 ((𝑣 ∈ Fin ∧ (𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))))
84 id 22 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥𝑦 = 𝑥)
85 oveq1 6617 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (𝑦(ball‘𝑀)(𝑐 / 2)) = (𝑥(ball‘𝑀)(𝑐 / 2)))
8685imaeq2d 5430 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) = (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2))))
8784, 86eleq12d 2692 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))
8887ralrab2 3358 . . . . . . . . . . . . 13 (∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))
8983, 88sylib 208 . . . . . . . . . . . 12 ((𝑣 ∈ Fin ∧ (𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))
9072, 75, 893jca 1240 . . . . . . . . . . 11 ((𝑣 ∈ Fin ∧ (𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2))))))
9190ex 450 . . . . . . . . . 10 (𝑣 ∈ Fin → ((𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))) → (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))))
9251, 91syl 17 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → ((𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))) → (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))))
93 simpr2 1066 . . . . . . . . . . . . 13 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑓:dom 𝑓𝑌)
94 frn 6015 . . . . . . . . . . . . 13 (𝑓:dom 𝑓𝑌 → ran 𝑓𝑌)
9593, 94syl 17 . . . . . . . . . . . 12 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ran 𝑓𝑌)
96 ffn 6007 . . . . . . . . . . . . . . 15 (𝑓:dom 𝑓𝑌𝑓 Fn dom 𝑓)
9793, 96syl 17 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑓 Fn dom 𝑓)
9851adantr 481 . . . . . . . . . . . . . . 15 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑣 ∈ Fin)
99 simpr1 1065 . . . . . . . . . . . . . . 15 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → dom 𝑓𝑣)
100 ssfi 8132 . . . . . . . . . . . . . . 15 ((𝑣 ∈ Fin ∧ dom 𝑓𝑣) → dom 𝑓 ∈ Fin)
10198, 99, 100syl2anc 692 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → dom 𝑓 ∈ Fin)
102 fnfi 8190 . . . . . . . . . . . . . 14 ((𝑓 Fn dom 𝑓 ∧ dom 𝑓 ∈ Fin) → 𝑓 ∈ Fin)
10397, 101, 102syl2anc 692 . . . . . . . . . . . . 13 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑓 ∈ Fin)
104 rnfi 8201 . . . . . . . . . . . . 13 (𝑓 ∈ Fin → ran 𝑓 ∈ Fin)
105103, 104syl 17 . . . . . . . . . . . 12 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ran 𝑓 ∈ Fin)
106 elfpw 8220 . . . . . . . . . . . 12 (ran 𝑓 ∈ (𝒫 𝑌 ∩ Fin) ↔ (ran 𝑓𝑌 ∧ ran 𝑓 ∈ Fin))
10795, 105, 106sylanbrc 697 . . . . . . . . . . 11 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ran 𝑓 ∈ (𝒫 𝑌 ∩ Fin))
108 oveq1 6617 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥(ball‘𝑁)𝑐) = (𝑧(ball‘𝑁)𝑐))
109108cbviunv 4530 . . . . . . . . . . . 12 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑁)𝑐) = 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐)
1103ad4antr 767 . . . . . . . . . . . . . . . . 17 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑧 ∈ ran 𝑓) → 𝑁 ∈ (Met‘𝑌))
111 metxmet 22062 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (Met‘𝑌) → 𝑁 ∈ (∞Met‘𝑌))
112110, 111syl 17 . . . . . . . . . . . . . . . 16 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑧 ∈ ran 𝑓) → 𝑁 ∈ (∞Met‘𝑌))
11395sselda 3587 . . . . . . . . . . . . . . . 16 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑧 ∈ ran 𝑓) → 𝑧𝑌)
114 rpxr 11792 . . . . . . . . . . . . . . . . 17 (𝑐 ∈ ℝ+𝑐 ∈ ℝ*)
115114ad4antlr 768 . . . . . . . . . . . . . . . 16 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑧 ∈ ran 𝑓) → 𝑐 ∈ ℝ*)
116 blssm 22146 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑧𝑌𝑐 ∈ ℝ*) → (𝑧(ball‘𝑁)𝑐) ⊆ 𝑌)
117112, 113, 115, 116syl3anc 1323 . . . . . . . . . . . . . . 15 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑧 ∈ ran 𝑓) → (𝑧(ball‘𝑁)𝑐) ⊆ 𝑌)
118117ralrimiva 2961 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ∀𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐) ⊆ 𝑌)
119 iunss 4532 . . . . . . . . . . . . . 14 ( 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐) ⊆ 𝑌 ↔ ∀𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐) ⊆ 𝑌)
120118, 119sylibr 224 . . . . . . . . . . . . 13 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐) ⊆ 𝑌)
121 iunin1 4556 . . . . . . . . . . . . . . 15 𝑦𝑣 ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = ( 𝑦𝑣 (𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌)
122 simplrr 800 . . . . . . . . . . . . . . . . 17 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))
12355cbviunv 4530 . . . . . . . . . . . . . . . . 17 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)) = 𝑦𝑣 (𝑦(ball‘𝑀)(𝑐 / 2))
124122, 123syl6sseq 3635 . . . . . . . . . . . . . . . 16 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑌 𝑦𝑣 (𝑦(ball‘𝑀)(𝑐 / 2)))
125 sseqin2 3800 . . . . . . . . . . . . . . . 16 (𝑌 𝑦𝑣 (𝑦(ball‘𝑀)(𝑐 / 2)) ↔ ( 𝑦𝑣 (𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = 𝑌)
126124, 125sylib 208 . . . . . . . . . . . . . . 15 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ( 𝑦𝑣 (𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = 𝑌)
127121, 126syl5eq 2667 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑦𝑣 ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = 𝑌)
128 0ss 3949 . . . . . . . . . . . . . . . . . . 19 ∅ ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐)
129 sseq1 3610 . . . . . . . . . . . . . . . . . . 19 (((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = ∅ → (((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐) ↔ ∅ ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐)))
130128, 129mpbiri 248 . . . . . . . . . . . . . . . . . 18 (((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = ∅ → ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
131130a1i 11 . . . . . . . . . . . . . . . . 17 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦𝑣) → (((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = ∅ → ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐)))
132 simpr3 1067 . . . . . . . . . . . . . . . . . . 19 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))
13356neeq1d 2849 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑦 → (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ ↔ ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅))
134 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑦𝑥 = 𝑦)
13555imaeq2d 5430 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑦 → (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2))) = (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))))
136134, 135eleq12d 2692 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑦 → (𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2))) ↔ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))))
137133, 136imbi12d 334 . . . . . . . . . . . . . . . . . . . 20 (𝑥 = 𝑦 → ((((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))) ↔ (((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))))))
138137rspccva 3297 . . . . . . . . . . . . . . . . . . 19 ((∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ 𝑦𝑣) → (((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))))
139132, 138sylan 488 . . . . . . . . . . . . . . . . . 18 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦𝑣) → (((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))))
14015ad5antr 769 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → 𝑀 ∈ (∞Met‘𝑋))
141 cnvimass 5449 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ⊆ dom 𝑓
14249simplbi 476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣𝑋)
143142ad2antrl 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → 𝑣𝑋)
144143adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑣𝑋)
14599, 144sstrd 3597 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → dom 𝑓𝑋)
146141, 145syl5ss 3598 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ⊆ 𝑋)
147146sselda 3587 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → 𝑦𝑋)
148 simp-4r 806 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → 𝑐 ∈ ℝ+)
149148rpred 11824 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → 𝑐 ∈ ℝ)
150 elpreima 6298 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 Fn dom 𝑓 → (𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ (𝑦 ∈ dom 𝑓 ∧ (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))))
151150simplbda 653 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 Fn dom 𝑓𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))
15297, 151sylan 488 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))
153 blhalf 22133 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑐 ∈ ℝ ∧ (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑦(ball‘𝑀)(𝑐 / 2)) ⊆ ((𝑓𝑦)(ball‘𝑀)𝑐))
154140, 147, 149, 152, 153syl22anc 1324 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑦(ball‘𝑀)(𝑐 / 2)) ⊆ ((𝑓𝑦)(ball‘𝑀)𝑐))
155 ssrin 3821 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑦(ball‘𝑀)(𝑐 / 2)) ⊆ ((𝑓𝑦)(ball‘𝑀)𝑐) → ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ (((𝑓𝑦)(ball‘𝑀)𝑐) ∩ 𝑌))
156154, 155syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ (((𝑓𝑦)(ball‘𝑀)𝑐) ∩ 𝑌))
157141sseli 3583 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) → 𝑦 ∈ dom 𝑓)
158 ffvelrn 6318 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓:dom 𝑓𝑌𝑦 ∈ dom 𝑓) → (𝑓𝑦) ∈ 𝑌)
15993, 157, 158syl2an 494 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑓𝑦) ∈ 𝑌)
160 simp-5r 808 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → 𝑌𝑋)
161160, 22sylib 208 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑋𝑌) = 𝑌)
162159, 161eleqtrrd 2701 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑓𝑦) ∈ (𝑋𝑌))
163114ad4antlr 768 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → 𝑐 ∈ ℝ*)
1641blres 22159 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑓𝑦) ∈ (𝑋𝑌) ∧ 𝑐 ∈ ℝ*) → ((𝑓𝑦)(ball‘𝑁)𝑐) = (((𝑓𝑦)(ball‘𝑀)𝑐) ∩ 𝑌))
165140, 162, 163, 164syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → ((𝑓𝑦)(ball‘𝑁)𝑐) = (((𝑓𝑦)(ball‘𝑀)𝑐) ∩ 𝑌))
166156, 165sseqtr4d 3626 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ ((𝑓𝑦)(ball‘𝑁)𝑐))
167 fnfvelrn 6317 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓 Fn dom 𝑓𝑦 ∈ dom 𝑓) → (𝑓𝑦) ∈ ran 𝑓)
16897, 157, 167syl2an 494 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑓𝑦) ∈ ran 𝑓)
169 oveq1 6617 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = (𝑓𝑦) → (𝑧(ball‘𝑁)𝑐) = ((𝑓𝑦)(ball‘𝑁)𝑐))
170169ssiun2s 4535 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓𝑦) ∈ ran 𝑓 → ((𝑓𝑦)(ball‘𝑁)𝑐) ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
171168, 170syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → ((𝑓𝑦)(ball‘𝑁)𝑐) ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
172166, 171sstrd 3597 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
173172adantlr 750 . . . . . . . . . . . . . . . . . . 19 (((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦𝑣) ∧ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
174173ex 450 . . . . . . . . . . . . . . . . . 18 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦𝑣) → (𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) → ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐)))
175139, 174syld 47 . . . . . . . . . . . . . . . . 17 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦𝑣) → (((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐)))
176131, 175pm2.61dne 2876 . . . . . . . . . . . . . . . 16 ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦𝑣) → ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
177176ralrimiva 2961 . . . . . . . . . . . . . . 15 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ∀𝑦𝑣 ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
178 iunss 4532 . . . . . . . . . . . . . . 15 ( 𝑦𝑣 ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐) ↔ ∀𝑦𝑣 ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
179177, 178sylibr 224 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑦𝑣 ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
180127, 179eqsstr3d 3624 . . . . . . . . . . . . 13 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑌 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
181120, 180eqssd 3604 . . . . . . . . . . . 12 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐) = 𝑌)
182109, 181syl5eq 2667 . . . . . . . . . . 11 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑁)𝑐) = 𝑌)
183 iuneq1 4505 . . . . . . . . . . . . 13 (𝑤 = ran 𝑓 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑁)𝑐))
184183eqeq1d 2623 . . . . . . . . . . . 12 (𝑤 = ran 𝑓 → ( 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑁)𝑐) = 𝑌))
185184rspcev 3298 . . . . . . . . . . 11 ((ran 𝑓 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑁)𝑐) = 𝑌) → ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌)
186107, 182, 185syl2anc 692 . . . . . . . . . 10 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌)
187186ex 450 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → ((dom 𝑓𝑣𝑓:dom 𝑓𝑌 ∧ ∀𝑥𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2))))) → ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
18892, 187syld 47 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → ((𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))) → ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
189188exlimdv 1858 . . . . . . 7 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → (∃𝑓(𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))) → ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
19069, 189mpd 15 . . . . . 6 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌)
191190rexlimdvaa 3026 . . . . 5 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)) → ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
19248, 191syld 47 . . . 4 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) → (∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) → ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
193192ralrimdva 2964 . . 3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) → ∀𝑐 ∈ ℝ+𝑤 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
194 istotbnd3 33237 . . . . 5 (𝑁 ∈ (TotBnd‘𝑌) ↔ (𝑁 ∈ (Met‘𝑌) ∧ ∀𝑐 ∈ ℝ+𝑤 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
195194baib 943 . . . 4 (𝑁 ∈ (Met‘𝑌) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑐 ∈ ℝ+𝑤 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
1963, 195syl 17 . . 3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑐 ∈ ℝ+𝑤 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
197193, 196sylibrd 249 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) → 𝑁 ∈ (TotBnd‘𝑌)))
19840, 197impbid 202 1 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135   ciun 4490   × cxp 5077  ccnv 5078  dom cdm 5079  ran crn 5080  cres 5081  cima 5082   Fn wfn 5847  wf 5848  cfv 5852  (class class class)co 6610  Fincfn 7907  cr 9887  *cxr 10025   / cdiv 10636  2c2 11022  +crp 11784  ∞Metcxmt 19663  Metcme 19664  ballcbl 19665  TotBndctotbnd 33232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9944  ax-resscn 9945  ax-1cn 9946  ax-icn 9947  ax-addcl 9948  ax-addrcl 9949  ax-mulcl 9950  ax-mulrcl 9951  ax-mulcom 9952  ax-addass 9953  ax-mulass 9954  ax-distr 9955  ax-i2m1 9956  ax-1ne0 9957  ax-1rid 9958  ax-rnegex 9959  ax-rrecex 9960  ax-cnre 9961  ax-pre-lttri 9962  ax-pre-lttrn 9963  ax-pre-ltadd 9964  ax-pre-mulgt0 9965
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7908  df-dom 7909  df-sdom 7910  df-fin 7911  df-pnf 10028  df-mnf 10029  df-xr 10030  df-ltxr 10031  df-le 10032  df-sub 10220  df-neg 10221  df-div 10637  df-2 11031  df-rp 11785  df-xneg 11898  df-xadd 11899  df-xmul 11900  df-psmet 19670  df-xmet 19671  df-met 19672  df-bl 19673  df-totbnd 33234
This theorem is referenced by:  sstotbnd  33241  sstotbnd3  33242
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