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Theorem sstotbnd2 33905
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 22388 . . . . 5 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑀 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌))
31, 2syl5eqel 2854 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → 𝑁 ∈ (Met‘𝑌))
4 istotbnd3 33902 . . . . 5 (𝑁 ∈ (TotBnd‘𝑌) ↔ (𝑁 ∈ (Met‘𝑌) ∧ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌))
54baib 525 . . . 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 760 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → 𝑌𝑋)
8 sspwb 5045 . . . . . . . . . 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)
109ssrind 3988 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → (𝒫 𝑌 ∩ Fin) ⊆ (𝒫 𝑋 ∩ Fin))
11 simprl 754 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → 𝑣 ∈ (𝒫 𝑌 ∩ Fin))
1210, 11sseldd 3753 . . . . . . 7 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → 𝑣 ∈ (𝒫 𝑋 ∩ Fin))
13 simprr 756 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)
14 metxmet 22359 . . . . . . . . . . . . . 14 (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
1514ad4antr 712 . . . . . . . . . . . . 13 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → 𝑀 ∈ (∞Met‘𝑋))
16 elfpw 8424 . . . . . . . . . . . . . . . . 17 (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ↔ (𝑣𝑌𝑣 ∈ Fin))
1716simplbi 485 . . . . . . . . . . . . . . . 16 (𝑣 ∈ (𝒫 𝑌 ∩ Fin) → 𝑣𝑌)
1817adantl 467 . . . . . . . . . . . . . . 15 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) → 𝑣𝑌)
1918sselda 3752 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → 𝑥𝑌)
20 simp-4r 770 . . . . . . . . . . . . . . 15 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → 𝑌𝑋)
21 sseqin2 3968 . . . . . . . . . . . . . . 15 (𝑌𝑋 ↔ (𝑋𝑌) = 𝑌)
2220, 21sylib 208 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → (𝑋𝑌) = 𝑌)
2319, 22eleqtrrd 2853 . . . . . . . . . . . . 13 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → 𝑥 ∈ (𝑋𝑌))
24 simpllr 760 . . . . . . . . . . . . . 14 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → 𝑑 ∈ ℝ+)
2524rpxrd 12076 . . . . . . . . . . . . 13 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → 𝑑 ∈ ℝ*)
261blres 22456 . . . . . . . . . . . . 13 ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ (𝑋𝑌) ∧ 𝑑 ∈ ℝ*) → (𝑥(ball‘𝑁)𝑑) = ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌))
2715, 23, 25, 26syl3anc 1476 . . . . . . . . . . . 12 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → (𝑥(ball‘𝑁)𝑑) = ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌))
28 inss1 3981 . . . . . . . . . . . 12 ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ⊆ (𝑥(ball‘𝑀)𝑑)
2927, 28syl6eqss 3804 . . . . . . . . . . 11 (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥𝑣) → (𝑥(ball‘𝑁)𝑑) ⊆ (𝑥(ball‘𝑀)𝑑))
3029ralrimiva 3115 . . . . . . . . . 10 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) → ∀𝑥𝑣 (𝑥(ball‘𝑁)𝑑) ⊆ (𝑥(ball‘𝑀)𝑑))
31 ss2iun 4670 . . . . . . . . . 10 (∀𝑥𝑣 (𝑥(ball‘𝑁)𝑑) ⊆ (𝑥(ball‘𝑀)𝑑) → 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) ⊆ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑))
3230, 31syl 17 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) → 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) ⊆ 𝑥𝑣 (𝑥(ball‘𝑀)𝑑))
3332adantrr 696 . . . . . . . 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))
3413, 33eqsstr3d 3789 . . . . . . 7 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑑 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑))
3512, 34jca 501 . . . . . 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)))
3635ex 397 . . . . 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))))
3736reximdv2 3162 . . . 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)))
3837ralimdva 3111 . . 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)))
396, 38sylbid 230 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) → ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑)))
40 simpr 471 . . . . . . 7 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) → 𝑐 ∈ ℝ+)
4140rphalfcld 12087 . . . . . 6 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) → (𝑐 / 2) ∈ ℝ+)
42 oveq2 6801 . . . . . . . . . 10 (𝑑 = (𝑐 / 2) → (𝑥(ball‘𝑀)𝑑) = (𝑥(ball‘𝑀)(𝑐 / 2)))
4342iuneq2d 4681 . . . . . . . . 9 (𝑑 = (𝑐 / 2) → 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) = 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))
4443sseq2d 3782 . . . . . . . 8 (𝑑 = (𝑐 / 2) → (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ↔ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2))))
4544rexbidv 3200 . . . . . . 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))))
4645rspcv 3456 . . . . . 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))))
4741, 46syl 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))))
48 elfpw 8424 . . . . . . . . . . 11 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣𝑋𝑣 ∈ Fin))
4948simprbi 484 . . . . . . . . . 10 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ∈ Fin)
5049ad2antrl 707 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → 𝑣 ∈ Fin)
51 ssrab2 3836 . . . . . . . . 9 {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ⊆ 𝑣
52 ssfi 8336 . . . . . . . . 9 ((𝑣 ∈ Fin ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ⊆ 𝑣) → {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ∈ Fin)
5350, 51, 52sylancl 574 . . . . . . . 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)
54 oveq1 6800 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → (𝑥(ball‘𝑀)(𝑐 / 2)) = (𝑦(ball‘𝑀)(𝑐 / 2)))
5554ineq1d 3964 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌))
56 incom 3956 . . . . . . . . . . . . . . 15 ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = (𝑌 ∩ (𝑦(ball‘𝑀)(𝑐 / 2)))
5755, 56syl6eq 2821 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = (𝑌 ∩ (𝑦(ball‘𝑀)(𝑐 / 2))))
58 dfin5 3731 . . . . . . . . . . . . . 14 (𝑌 ∩ (𝑦(ball‘𝑀)(𝑐 / 2))) = {𝑧𝑌𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))}
5957, 58syl6eq 2821 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = {𝑧𝑌𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))})
6059neeq1d 3002 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ ↔ {𝑧𝑌𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))} ≠ ∅))
61 rabn0 4104 . . . . . . . . . . . 12 ({𝑧𝑌𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))} ≠ ∅ ↔ ∃𝑧𝑌 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))
6260, 61syl6bb 276 . . . . . . . . . . 11 (𝑥 = 𝑦 → (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ ↔ ∃𝑧𝑌 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
6362elrab 3515 . . . . . . . . . 10 (𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ↔ (𝑦𝑣 ∧ ∃𝑧𝑌 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
6463simprbi 484 . . . . . . . . 9 (𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} → ∃𝑧𝑌 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))
6564rgen 3071 . . . . . . . 8 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}∃𝑧𝑌 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))
66 eleq1 2838 . . . . . . . . 9 (𝑧 = (𝑓𝑦) → (𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2)) ↔ (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
6766ac6sfi 8360 . . . . . . . 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))))
6853, 65, 67sylancl 574 . . . . . . 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))))
69 fdm 6191 . . . . . . . . . . . . . 14 (𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 → dom 𝑓 = {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅})
7069ad2antrl 707 . . . . . . . . . . . . 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) ≠ ∅})
7170, 51syl6eqss 3804 . . . . . . . . . . . 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)
72 simprl 754 . . . . . . . . . . . . 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)
7370feq2d 6171 . . . . . . . . . . . . 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))
7472, 73mpbird 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)
75 simprr 756 . . . . . . . . . . . . . 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)))
76 ffn 6185 . . . . . . . . . . . . . . . . . 18 (𝑓:{𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌𝑓 Fn {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅})
77 elpreima 6480 . . . . . . . . . . . . . . . . . 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)))))
7876, 77syl 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)))))
7978baibd 529 . . . . . . . . . . . . . . . 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))))
8079ralbidva 3134 . . . . . . . . . . . . . . 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))))
8180ad2antrl 707 . . . . . . . . . . . . . 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))))
8275, 81mpbird 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))))
83 id 22 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥𝑦 = 𝑥)
84 oveq1 6800 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → (𝑦(ball‘𝑀)(𝑐 / 2)) = (𝑥(ball‘𝑀)(𝑐 / 2)))
8584imaeq2d 5607 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) = (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2))))
8683, 85eleq12d 2844 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ 𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))
8786ralrab2 3524 . . . . . . . . . . . . 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)))))
8882, 87sylib 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)))))
8971, 74, 883jca 1122 . . . . . . . . . . 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))))))
9089ex 397 . . . . . . . . . 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)))))))
9150, 90syl 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)))))))
92 simpr2 1235 . . . . . . . . . . . . 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)
93 frn 6193 . . . . . . . . . . . . 13 (𝑓:dom 𝑓𝑌 → ran 𝑓𝑌)
9492, 93syl 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)
95 ffn 6185 . . . . . . . . . . . . . . 15 (𝑓:dom 𝑓𝑌𝑓 Fn dom 𝑓)
9692, 95syl 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)
9750adantr 466 . . . . . . . . . . . . . . 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)
98 simpr1 1233 . . . . . . . . . . . . . . 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)
99 ssfi 8336 . . . . . . . . . . . . . . 15 ((𝑣 ∈ Fin ∧ dom 𝑓𝑣) → dom 𝑓 ∈ Fin)
10097, 98, 99syl2anc 573 . . . . . . . . . . . . . 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)
101 fnfi 8394 . . . . . . . . . . . . . 14 ((𝑓 Fn dom 𝑓 ∧ dom 𝑓 ∈ Fin) → 𝑓 ∈ Fin)
10296, 100, 101syl2anc 573 . . . . . . . . . . . . 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)
103 rnfi 8405 . . . . . . . . . . . . 13 (𝑓 ∈ Fin → ran 𝑓 ∈ Fin)
104102, 103syl 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)
105 elfpw 8424 . . . . . . . . . . . 12 (ran 𝑓 ∈ (𝒫 𝑌 ∩ Fin) ↔ (ran 𝑓𝑌 ∧ ran 𝑓 ∈ Fin))
10694, 104, 105sylanbrc 572 . . . . . . . . . . 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))
107 oveq1 6800 . . . . . . . . . . . . 13 (𝑥 = 𝑧 → (𝑥(ball‘𝑁)𝑐) = (𝑧(ball‘𝑁)𝑐))
108107cbviunv 4693 . . . . . . . . . . . 12 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑁)𝑐) = 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐)
1093ad4antr 712 . . . . . . . . . . . . . . . . 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))
110 metxmet 22359 . . . . . . . . . . . . . . . . 17 (𝑁 ∈ (Met‘𝑌) → 𝑁 ∈ (∞Met‘𝑌))
111109, 110syl 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))
11294sselda 3752 . . . . . . . . . . . . . . . 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)
113 rpxr 12043 . . . . . . . . . . . . . . . . 17 (𝑐 ∈ ℝ+𝑐 ∈ ℝ*)
114113ad4antlr 714 . . . . . . . . . . . . . . . 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*)
115 blssm 22443 . . . . . . . . . . . . . . . 16 ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑧𝑌𝑐 ∈ ℝ*) → (𝑧(ball‘𝑁)𝑐) ⊆ 𝑌)
116111, 112, 114, 115syl3anc 1476 . . . . . . . . . . . . . . 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)
117116ralrimiva 3115 . . . . . . . . . . . . . 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)
118 iunss 4695 . . . . . . . . . . . . . 14 ( 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐) ⊆ 𝑌 ↔ ∀𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐) ⊆ 𝑌)
119117, 118sylibr 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)
120 iunin1 4719 . . . . . . . . . . . . . . 15 𝑦𝑣 ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = ( 𝑦𝑣 (𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌)
121 simplrr 763 . . . . . . . . . . . . . . . . 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)))
12254cbviunv 4693 . . . . . . . . . . . . . . . . 17 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)) = 𝑦𝑣 (𝑦(ball‘𝑀)(𝑐 / 2))
123121, 122syl6sseq 3800 . . . . . . . . . . . . . . . 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)))
124 sseqin2 3968 . . . . . . . . . . . . . . . 16 (𝑌 𝑦𝑣 (𝑦(ball‘𝑀)(𝑐 / 2)) ↔ ( 𝑦𝑣 (𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = 𝑌)
125123, 124sylib 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)
126120, 125syl5eq 2817 . . . . . . . . . . . . . 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)
127 0ss 4116 . . . . . . . . . . . . . . . . . . 19 ∅ ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐)
128 sseq1 3775 . . . . . . . . . . . . . . . . . . 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)))
129127, 128mpbiri 248 . . . . . . . . . . . . . . . . . 18 (((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = ∅ → ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
130129a1i 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)))
131 simpr3 1237 . . . . . . . . . . . . . . . . . . 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)))))
13255neeq1d 3002 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑦 → (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ ↔ ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅))
133 id 22 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑦𝑥 = 𝑦)
13454imaeq2d 5607 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑦 → (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2))) = (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))))
135133, 134eleq12d 2844 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 = 𝑦 → (𝑥 ∈ (𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2))) ↔ 𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))))
136132, 135imbi12d 333 . . . . . . . . . . . . . . . . . . . 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))))))
137136rspccva 3459 . . . . . . . . . . . . . . . . . . 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)))))
138131, 137sylan 569 . . . . . . . . . . . . . . . . . 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)))))
13914ad5antr 716 . . . . . . . . . . . . . . . . . . . . . . . 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))
140 cnvimass 5626 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ⊆ dom 𝑓
14148simplbi 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣𝑋)
142141ad2antrl 707 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑐 ∈ ℝ+) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → 𝑣𝑋)
143142adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
14498, 143sstrd 3762 . . . . . . . . . . . . . . . . . . . . . . . . . 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)
145140, 144syl5ss 3763 . . . . . . . . . . . . . . . . . . . . . . . . 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)
146145sselda 3752 . . . . . . . . . . . . . . . . . . . . . . . 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)
147 simp-4r 770 . . . . . . . . . . . . . . . . . . . . . . . . 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+)
148147rpred 12075 . . . . . . . . . . . . . . . . . . . . . . . 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)
149 elpreima 6480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 Fn dom 𝑓 → (𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ (𝑦 ∈ dom 𝑓 ∧ (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))))
150149simplbda 487 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓 Fn dom 𝑓𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))
15196, 150sylan 569 . . . . . . . . . . . . . . . . . . . . . . . 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)))
152 blhalf 22430 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋) ∧ (𝑐 ∈ ℝ ∧ (𝑓𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑦(ball‘𝑀)(𝑐 / 2)) ⊆ ((𝑓𝑦)(ball‘𝑀)𝑐))
153139, 146, 148, 151, 152syl22anc 1477 . . . . . . . . . . . . . . . . . . . . . . 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))
154153ssrind 3988 . . . . . . . . . . . . . . . . . . . . . 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))
155140sseli 3748 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ (𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) → 𝑦 ∈ dom 𝑓)
156 ffvelrn 6500 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓:dom 𝑓𝑌𝑦 ∈ dom 𝑓) → (𝑓𝑦) ∈ 𝑌)
15792, 155, 156syl2an 583 . . . . . . . . . . . . . . . . . . . . . . . 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)
158 simp-5r 774 . . . . . . . . . . . . . . . . . . . . . . . . 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)
159158, 21sylib 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)
160157, 159eleqtrrd 2853 . . . . . . . . . . . . . . . . . . . . . . 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))
161113ad4antlr 714 . . . . . . . . . . . . . . . . . . . . . . 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*)
1621blres 22456 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑓𝑦) ∈ (𝑋𝑌) ∧ 𝑐 ∈ ℝ*) → ((𝑓𝑦)(ball‘𝑁)𝑐) = (((𝑓𝑦)(ball‘𝑀)𝑐) ∩ 𝑌))
163139, 160, 161, 162syl3anc 1476 . . . . . . . . . . . . . . . . . . . . . 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))
164154, 163sseqtr4d 3791 . . . . . . . . . . . . . . . . . . . . 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))
165 fnfvelrn 6499 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓 Fn dom 𝑓𝑦 ∈ dom 𝑓) → (𝑓𝑦) ∈ ran 𝑓)
16696, 155, 165syl2an 583 . . . . . . . . . . . . . . . . . . . . . 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)
167 oveq1 6800 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 = (𝑓𝑦) → (𝑧(ball‘𝑁)𝑐) = ((𝑓𝑦)(ball‘𝑁)𝑐))
168167ssiun2s 4698 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑓𝑦) ∈ ran 𝑓 → ((𝑓𝑦)(ball‘𝑁)𝑐) ⊆ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
169166, 168syl 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))
170164, 169sstrd 3762 . . . . . . . . . . . . . . . . . . . 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))
171170adantlr 694 . . . . . . . . . . . . . . . . . . 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))
172171ex 397 . . . . . . . . . . . . . . . . . 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)))
173138, 172syld 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)))
174130, 173pm2.61dne 3029 . . . . . . . . . . . . . . . 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))
175174ralrimiva 3115 . . . . . . . . . . . . . . 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))
176 iunss 4695 . . . . . . . . . . . . . . 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))
177175, 176sylibr 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))
178126, 177eqsstr3d 3789 . . . . . . . . . . . . 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))
179119, 178eqssd 3769 . . . . . . . . . . . 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)
180108, 179syl5eq 2817 . . . . . . . . . . 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)
181 iuneq1 4668 . . . . . . . . . . . . 13 (𝑤 = ran 𝑓 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑁)𝑐))
182181eqeq1d 2773 . . . . . . . . . . . 12 (𝑤 = ran 𝑓 → ( 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑁)𝑐) = 𝑌))
183182rspcev 3460 . . . . . . . . . . 11 ((ran 𝑓 ∈ (𝒫 𝑌 ∩ Fin) ∧ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑁)𝑐) = 𝑌) → ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌)
184106, 180, 183syl2anc 573 . . . . . . . . . 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)
185184ex 397 . . . . . . . . 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))
18691, 185syld 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))
187186exlimdv 2013 . . . . . . 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))
18868, 187mpd 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)
189188rexlimdvaa 3180 . . . . 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))
19047, 189syld 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))
191190ralrimdva 3118 . . 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))
192 istotbnd3 33902 . . . . 5 (𝑁 ∈ (TotBnd‘𝑌) ↔ (𝑁 ∈ (Met‘𝑌) ∧ ∀𝑐 ∈ ℝ+𝑤 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
193192baib 525 . . . 4 (𝑁 ∈ (Met‘𝑌) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑐 ∈ ℝ+𝑤 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
1943, 193syl 17 . . 3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑐 ∈ ℝ+𝑤 ∈ (𝒫 𝑌 ∩ Fin) 𝑥𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
195191, 194sylibrd 249 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) → 𝑁 ∈ (TotBnd‘𝑌)))
19639, 195impbid 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 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  wne 2943  wral 3061  wrex 3062  {crab 3065  cin 3722  wss 3723  c0 4063  𝒫 cpw 4297   ciun 4654   × cxp 5247  ccnv 5248  dom cdm 5249  ran crn 5250  cres 5251  cima 5252   Fn wfn 6026  wf 6027  cfv 6031  (class class class)co 6793  Fincfn 8109  cr 10137  *cxr 10275   / cdiv 10886  2c2 11272  +crp 12035  ∞Metcxmt 19946  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  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215
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-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  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-riota 6754  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-map 8011  df-en 8110  df-dom 8111  df-sdom 8112  df-fin 8113  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-div 10887  df-2 11281  df-rp 12036  df-xneg 12151  df-xadd 12152  df-xmul 12153  df-psmet 19953  df-xmet 19954  df-met 19955  df-bl 19956  df-totbnd 33899
This theorem is referenced by:  sstotbnd  33906  sstotbnd3  33907
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