sstotbnd2 - Mathbox for Jeff Madsen

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Theorem sstotbnd2 33879

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) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (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.

Step	Hyp	Ref	Expression
1		sstotbnd.2	. . . . 5 ⊢ 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))
2		metres2 22377	. . . . 5 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑀 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌))
3	1, 2	syl5eqel 2889	. . . 4 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → 𝑁 ∈ (Met‘𝑌))
4		istotbnd3 33876	. . . . 5 ⊢ (𝑁 ∈ (TotBnd‘𝑌) ↔ (𝑁 ∈ (Met‘𝑌) ∧ ∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ (𝒫 𝑌 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌))
5	4	baib 527	. . . 4 ⊢ (𝑁 ∈ (Met‘𝑌) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ (𝒫 𝑌 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌))
6	3, 5	syl 17	. . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ (𝒫 𝑌 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌))
7		simpllr 784	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → 𝑌 ⊆ 𝑋)
8		sspwb 5107	. . . . . . . . . 10 ⊢ (𝑌 ⊆ 𝑋 ↔ 𝒫 𝑌 ⊆ 𝒫 𝑋)
9	7, 8	sylib 209	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → 𝒫 𝑌 ⊆ 𝒫 𝑋)
10	9	ssrind 4036	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → (𝒫 𝑌 ∩ Fin) ⊆ (𝒫 𝑋 ∩ Fin))
11		simprl 778	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → 𝑣 ∈ (𝒫 𝑌 ∩ Fin))
12	10, 11	sseldd 3799	. . . . . . 7 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → 𝑣 ∈ (𝒫 𝑋 ∩ Fin))
13		simprr 780	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)
14		metxmet 22348	. . . . . . . . . . . . . 14 ⊢ (𝑀 ∈ (Met‘𝑋) → 𝑀 ∈ (∞Met‘𝑋))
15	14	ad4antr 715	. . . . . . . . . . . . 13 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥 ∈ 𝑣) → 𝑀 ∈ (∞Met‘𝑋))
16		elfpw 8503	. . . . . . . . . . . . . . . . 17 ⊢ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ↔ (𝑣 ⊆ 𝑌 ∧ 𝑣 ∈ Fin))
17	16	simplbi 487	. . . . . . . . . . . . . . . 16 ⊢ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) → 𝑣 ⊆ 𝑌)
18	17	adantl 469	. . . . . . . . . . . . . . 15 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) → 𝑣 ⊆ 𝑌)
19	18	sselda 3798	. . . . . . . . . . . . . 14 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥 ∈ 𝑣) → 𝑥 ∈ 𝑌)
20		simp-4r 794	. . . . . . . . . . . . . . 15 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥 ∈ 𝑣) → 𝑌 ⊆ 𝑋)
21		sseqin2 4016	. . . . . . . . . . . . . . 15 ⊢ (𝑌 ⊆ 𝑋 ↔ (𝑋 ∩ 𝑌) = 𝑌)
22	20, 21	sylib 209	. . . . . . . . . . . . . 14 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥 ∈ 𝑣) → (𝑋 ∩ 𝑌) = 𝑌)
23	19, 22	eleqtrrd 2888	. . . . . . . . . . . . 13 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥 ∈ 𝑣) → 𝑥 ∈ (𝑋 ∩ 𝑌))
24		simpllr 784	. . . . . . . . . . . . . 14 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥 ∈ 𝑣) → 𝑑 ∈ ℝ⁺)
25	24	rpxrd 12083	. . . . . . . . . . . . 13 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥 ∈ 𝑣) → 𝑑 ∈ ℝ^*)
26	1	blres 22445	. . . . . . . . . . . . 13 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ (𝑋 ∩ 𝑌) ∧ 𝑑 ∈ ℝ^*) → (𝑥(ball‘𝑁)𝑑) = ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌))
27	15, 23, 25, 26	syl3anc 1483	. . . . . . . . . . . 12 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥 ∈ 𝑣) → (𝑥(ball‘𝑁)𝑑) = ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌))
28		inss1 4029	. . . . . . . . . . . 12 ⊢ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ⊆ (𝑥(ball‘𝑀)𝑑)
29	27, 28	syl6eqss 3852	. . . . . . . . . . 11 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) ∧ 𝑥 ∈ 𝑣) → (𝑥(ball‘𝑁)𝑑) ⊆ (𝑥(ball‘𝑀)𝑑))
30	29	ralrimiva 3154	. . . . . . . . . 10 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) → ∀𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) ⊆ (𝑥(ball‘𝑀)𝑑))
31		ss2iun 4728	. . . . . . . . . 10 ⊢ (∀𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) ⊆ (𝑥(ball‘𝑀)𝑑) → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑))
32	30, 31	syl 17	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ 𝑣 ∈ (𝒫 𝑌 ∩ Fin)) → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑))
33	32	adantrr 699	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑))
34	13, 33	eqsstr3d 3837	. . . . . . 7 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑))
35	12, 34	jca 503	. . . . . 6 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌)) → (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑)))
36	35	ex 399	. . . . 5 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) → ((𝑣 ∈ (𝒫 𝑌 ∩ Fin) ∧ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌) → (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑))))
37	36	reximdv2 3201	. . . 4 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑑 ∈ ℝ⁺) → (∃𝑣 ∈ (𝒫 𝑌 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌 → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑)))
38	37	ralimdva 3150	. . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ (𝒫 𝑌 ∩ Fin)∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑁)𝑑) = 𝑌 → ∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑)))
39	6, 38	sylbid 231	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) → ∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑)))
40		simpr 473	. . . . . . 7 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) → 𝑐 ∈ ℝ⁺)
41	40	rphalfcld 12094	. . . . . 6 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) → (𝑐 / 2) ∈ ℝ⁺)
42		oveq2 6878	. . . . . . . . . 10 ⊢ (𝑑 = (𝑐 / 2) → (𝑥(ball‘𝑀)𝑑) = (𝑥(ball‘𝑀)(𝑐 / 2)))
43	42	iuneq2d 4739	. . . . . . . . 9 ⊢ (𝑑 = (𝑐 / 2) → ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) = ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))
44	43	sseq2d 3830	. . . . . . . 8 ⊢ (𝑑 = (𝑐 / 2) → (𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ↔ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2))))
45	44	rexbidv 3240	. . . . . . 7 ⊢ (𝑑 = (𝑐 / 2) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2))))
46	45	rspcv 3498	. . . . . 6 ⊢ ((𝑐 / 2) ∈ ℝ⁺ → (∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2))))
47	41, 46	syl 17	. . . . 5 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) → (∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) → ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2))))
48		elfpw 8503	. . . . . . . . . . 11 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 ⊆ 𝑋 ∧ 𝑣 ∈ Fin))
49	48	simprbi 486	. . . . . . . . . 10 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ∈ Fin)
50	49	ad2antrl 710	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → 𝑣 ∈ Fin)
51		ssrab2 3884	. . . . . . . . 9 ⊢ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ⊆ 𝑣
52		ssfi 8415	. . . . . . . . 9 ⊢ ((𝑣 ∈ Fin ∧ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ⊆ 𝑣) → {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ∈ Fin)
53	50, 51, 52	sylancl 576	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ∈ Fin)
54		oveq1 6877	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → (𝑥(ball‘𝑀)(𝑐 / 2)) = (𝑦(ball‘𝑀)(𝑐 / 2)))
55	54	ineq1d 4012	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌))
56		incom 4004	. . . . . . . . . . . . . . 15 ⊢ ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = (𝑌 ∩ (𝑦(ball‘𝑀)(𝑐 / 2)))
57	55, 56	syl6eq 2856	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = (𝑌 ∩ (𝑦(ball‘𝑀)(𝑐 / 2))))
58		dfin5 3777	. . . . . . . . . . . . . 14 ⊢ (𝑌 ∩ (𝑦(ball‘𝑀)(𝑐 / 2))) = {𝑧 ∈ 𝑌 ∣ 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))}
59	57, 58	syl6eq 2856	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = {𝑧 ∈ 𝑌 ∣ 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))})
60	59	neeq1d 3037	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ ↔ {𝑧 ∈ 𝑌 ∣ 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))} ≠ ∅))
61		rabn0 4158	. . . . . . . . . . . 12 ⊢ ({𝑧 ∈ 𝑌 ∣ 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))} ≠ ∅ ↔ ∃𝑧 ∈ 𝑌 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))
62	60, 61	syl6bb 278	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ ↔ ∃𝑧 ∈ 𝑌 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
63	62	elrab 3559	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ↔ (𝑦 ∈ 𝑣 ∧ ∃𝑧 ∈ 𝑌 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
64	63	simprbi 486	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} → ∃𝑧 ∈ 𝑌 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))
65	64	rgen 3110	. . . . . . . 8 ⊢ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}∃𝑧 ∈ 𝑌 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))
66		eleq1 2873	. . . . . . . . 9 ⊢ (𝑧 = (𝑓‘𝑦) → (𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2)) ↔ (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
67	66	ac6sfi 8439	. . . . . . . 8 ⊢ (({𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ∈ Fin ∧ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}∃𝑧 ∈ 𝑌 𝑧 ∈ (𝑦(ball‘𝑀)(𝑐 / 2))) → ∃𝑓(𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
68	53, 65, 67	sylancl 576	. . . . . . 7 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → ∃𝑓(𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
69		fdm 6260	. . . . . . . . . . . . . 14 ⊢ (𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 → dom 𝑓 = {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅})
70	69	ad2antrl 710	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ Fin ∧ (𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → dom 𝑓 = {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅})
71	70, 51	syl6eqss 3852	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ Fin ∧ (𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → dom 𝑓 ⊆ 𝑣)
72		simprl 778	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ Fin ∧ (𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → 𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌)
73	70	feq2d 6238	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ Fin ∧ (𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑓:dom 𝑓⟶𝑌 ↔ 𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌))
74	72, 73	mpbird 248	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ Fin ∧ (𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → 𝑓:dom 𝑓⟶𝑌)
75		simprr 780	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ Fin ∧ (𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))
76		ffn 6252	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 → 𝑓 Fn {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅})
77		elpreima 6555	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 Fn {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} → (𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ (𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ∧ (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))))
78	76, 77	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 → (𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ (𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} ∧ (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))))
79	78	baibd 531	. . . . . . . . . . . . . . . 16 ⊢ ((𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ 𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}) → (𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
80	79	ralbidva 3173	. . . . . . . . . . . . . . 15 ⊢ (𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 → (∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
81	80	ad2antrl 710	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ Fin ∧ (𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))))
82	75, 81	mpbird 248	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ Fin ∧ (𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))))
83		id 22	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → 𝑦 = 𝑥)
84		oveq1 6877	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑥 → (𝑦(ball‘𝑀)(𝑐 / 2)) = (𝑥(ball‘𝑀)(𝑐 / 2)))
85	84	imaeq2d 5676	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) = (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2))))
86	83, 85	eleq12d 2879	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))
87	86	ralrab2 3568	. . . . . . . . . . . . 13 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))
88	82, 87	sylib 209	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ Fin ∧ (𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))
89	71, 74, 88	3jca 1151	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ Fin ∧ (𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2))))))
90	89	ex 399	. . . . . . . . . 10 ⊢ (𝑣 ∈ Fin → ((𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))) → (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))))
91	50, 90	syl 17	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → ((𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))) → (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))))
92		simpr2 1243	. . . . . . . . . . . . 13 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑓:dom 𝑓⟶𝑌)
93	92	frnd 6259	. . . . . . . . . . . 12 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ran 𝑓 ⊆ 𝑌)
94	92	ffnd 6253	. . . . . . . . . . . . . 14 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑓 Fn dom 𝑓)
95	50	adantr 468	. . . . . . . . . . . . . . 15 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑣 ∈ Fin)
96		simpr1 1241	. . . . . . . . . . . . . . 15 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → dom 𝑓 ⊆ 𝑣)
97		ssfi 8415	. . . . . . . . . . . . . . 15 ⊢ ((𝑣 ∈ Fin ∧ dom 𝑓 ⊆ 𝑣) → dom 𝑓 ∈ Fin)
98	95, 96, 97	syl2anc 575	. . . . . . . . . . . . . 14 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → dom 𝑓 ∈ Fin)
99		fnfi 8473	. . . . . . . . . . . . . 14 ⊢ ((𝑓 Fn dom 𝑓 ∧ dom 𝑓 ∈ Fin) → 𝑓 ∈ Fin)
100	94, 98, 99	syl2anc 575	. . . . . . . . . . . . 13 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑓 ∈ Fin)
101		rnfi 8484	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ Fin → ran 𝑓 ∈ Fin)
102	100, 101	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ran 𝑓 ∈ Fin)
103		elfpw 8503	. . . . . . . . . . . 12 ⊢ (ran 𝑓 ∈ (𝒫 𝑌 ∩ Fin) ↔ (ran 𝑓 ⊆ 𝑌 ∧ ran 𝑓 ∈ Fin))
104	93, 102, 103	sylanbrc 574	. . . . . . . . . . 11 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ran 𝑓 ∈ (𝒫 𝑌 ∩ Fin))
105		oveq1 6877	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑧 → (𝑥(ball‘𝑁)𝑐) = (𝑧(ball‘𝑁)𝑐))
106	105	cbviunv 4751	. . . . . . . . . . . 12 ⊢ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑁)𝑐) = ∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐)
107	3	ad4antr 715	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑧 ∈ ran 𝑓) → 𝑁 ∈ (Met‘𝑌))
108		metxmet 22348	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (Met‘𝑌) → 𝑁 ∈ (∞Met‘𝑌))
109	107, 108	syl 17	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑧 ∈ ran 𝑓) → 𝑁 ∈ (∞Met‘𝑌))
110	93	sselda 3798	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑧 ∈ ran 𝑓) → 𝑧 ∈ 𝑌)
111		rpxr 12050	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 ∈ ℝ⁺ → 𝑐 ∈ ℝ^*)
112	111	ad4antlr 717	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑧 ∈ ran 𝑓) → 𝑐 ∈ ℝ^*)
113		blssm 22432	. . . . . . . . . . . . . . . 16 ⊢ ((𝑁 ∈ (∞Met‘𝑌) ∧ 𝑧 ∈ 𝑌 ∧ 𝑐 ∈ ℝ^*) → (𝑧(ball‘𝑁)𝑐) ⊆ 𝑌)
114	109, 110, 112, 113	syl3anc 1483	. . . . . . . . . . . . . . 15 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑧 ∈ ran 𝑓) → (𝑧(ball‘𝑁)𝑐) ⊆ 𝑌)
115	114	ralrimiva 3154	. . . . . . . . . . . . . 14 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ∀𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐) ⊆ 𝑌)
116		iunss 4753	. . . . . . . . . . . . . 14 ⊢ (∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐) ⊆ 𝑌 ↔ ∀𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐) ⊆ 𝑌)
117	115, 116	sylibr 225	. . . . . . . . . . . . 13 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐) ⊆ 𝑌)
118		iunin1 4777	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑦 ∈ 𝑣 ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = (∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌)
119		simplrr 787	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))
120	54	cbviunv 4751	. . . . . . . . . . . . . . . . 17 ⊢ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)) = ∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝑀)(𝑐 / 2))
121	119, 120	syl6sseq 3848	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑌 ⊆ ∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝑀)(𝑐 / 2)))
122		sseqin2 4016	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 ⊆ ∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝑀)(𝑐 / 2)) ↔ (∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = 𝑌)
123	121, 122	sylib 209	. . . . . . . . . . . . . . 15 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → (∪ 𝑦 ∈ 𝑣 (𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = 𝑌)
124	118, 123	syl5eq 2852	. . . . . . . . . . . . . 14 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ∪ 𝑦 ∈ 𝑣 ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = 𝑌)
125		0ss 4170	. . . . . . . . . . . . . . . . . . 19 ⊢ ∅ ⊆ ∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐)
126		sseq1 3823	. . . . . . . . . . . . . . . . . . 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)))
127	125, 126	mpbiri 249	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = ∅ → ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ ∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
128	127	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ 𝑣) → (((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) = ∅ → ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ ∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐)))
129		simpr3 1245	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))
130	55	neeq1d 3037	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑦 → (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ ↔ ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅))
131		id 22	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
132	54	imaeq2d 5676	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 = 𝑦 → (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2))) = (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))))
133	131, 132	eleq12d 2879	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2))) ↔ 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))))
134	130, 133	imbi12d 335	. . . . . . . . . . . . . . . . . . . 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))))))
135	134	rspccva 3501	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ 𝑦 ∈ 𝑣) → (((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))))
136	129, 135	sylan 571	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ 𝑣) → (((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))))
137	14	ad5antr 719	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → 𝑀 ∈ (∞Met‘𝑋))
138		cnvimass 5695	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ⊆ dom 𝑓
139	48	simplbi 487	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → 𝑣 ⊆ 𝑋)
140	139	ad2antrl 710	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → 𝑣 ⊆ 𝑋)
141	140	adantr 468	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑣 ⊆ 𝑋)
142	96, 141	sstrd 3808	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → dom 𝑓 ⊆ 𝑋)
143	138, 142	syl5ss 3809	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ⊆ 𝑋)
144	143	sselda 3798	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → 𝑦 ∈ 𝑋)
145		simp-4r 794	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → 𝑐 ∈ ℝ⁺)
146	145	rpred 12082	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → 𝑐 ∈ ℝ)
147		elpreima 6555	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 Fn dom 𝑓 → (𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) ↔ (𝑦 ∈ dom 𝑓 ∧ (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))))
148	147	simplbda 489	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓 Fn dom 𝑓 ∧ 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))
149	94, 148	sylan 571	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))
150		blhalf 22419	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑀 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ (𝑐 ∈ ℝ ∧ (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑦(ball‘𝑀)(𝑐 / 2)) ⊆ ((𝑓‘𝑦)(ball‘𝑀)𝑐))
151	137, 144, 146, 149, 150	syl22anc 858	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((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)) ⊆ ((𝑓‘𝑦)(ball‘𝑀)𝑐))
152	151	ssrind 4036	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((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) ⊆ (((𝑓‘𝑦)(ball‘𝑀)𝑐) ∩ 𝑌))
153	138	sseli 3794	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) → 𝑦 ∈ dom 𝑓)
154		ffvelrn 6575	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓:dom 𝑓⟶𝑌 ∧ 𝑦 ∈ dom 𝑓) → (𝑓‘𝑦) ∈ 𝑌)
155	92, 153, 154	syl2an 585	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑓‘𝑦) ∈ 𝑌)
156		simp-5r 798	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → 𝑌 ⊆ 𝑋)
157	156, 21	sylib 209	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑋 ∩ 𝑌) = 𝑌)
158	155, 157	eleqtrrd 2888	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑓‘𝑦) ∈ (𝑋 ∩ 𝑌))
159	111	ad4antlr 717	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → 𝑐 ∈ ℝ^*)
160	1	blres 22445	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑀 ∈ (∞Met‘𝑋) ∧ (𝑓‘𝑦) ∈ (𝑋 ∩ 𝑌) ∧ 𝑐 ∈ ℝ^*) → ((𝑓‘𝑦)(ball‘𝑁)𝑐) = (((𝑓‘𝑦)(ball‘𝑀)𝑐) ∩ 𝑌))
161	137, 158, 159, 160	syl3anc 1483	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → ((𝑓‘𝑦)(ball‘𝑁)𝑐) = (((𝑓‘𝑦)(ball‘𝑀)𝑐) ∩ 𝑌))
162	152, 161	sseqtr4d 3839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((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) ⊆ ((𝑓‘𝑦)(ball‘𝑁)𝑐))
163		fnfvelrn 6574	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓 Fn dom 𝑓 ∧ 𝑦 ∈ dom 𝑓) → (𝑓‘𝑦) ∈ ran 𝑓)
164	94, 153, 163	syl2an 585	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → (𝑓‘𝑦) ∈ ran 𝑓)
165		oveq1 6877	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑧 = (𝑓‘𝑦) → (𝑧(ball‘𝑁)𝑐) = ((𝑓‘𝑦)(ball‘𝑁)𝑐))
166	165	ssiun2s 4756	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑓‘𝑦) ∈ ran 𝑓 → ((𝑓‘𝑦)(ball‘𝑁)𝑐) ⊆ ∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
167	164, 166	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → ((𝑓‘𝑦)(ball‘𝑁)𝑐) ⊆ ∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
168	162, 167	sstrd 3808	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((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))
169	168	adantlr 697	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ 𝑣) ∧ 𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2)))) → ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ ∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
170	169	ex 399	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ 𝑣) → (𝑦 ∈ (^◡𝑓 “ (𝑦(ball‘𝑀)(𝑐 / 2))) → ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ ∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐)))
171	136, 170	syld 47	. . . . . . . . . . . . . . . . 17 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ 𝑣) → (((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ ∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐)))
172	128, 171	pm2.61dne 3064	. . . . . . . . . . . . . . . 16 ⊢ ((((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) ∧ 𝑦 ∈ 𝑣) → ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ ∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
173	172	ralrimiva 3154	. . . . . . . . . . . . . . 15 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ∀𝑦 ∈ 𝑣 ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ ∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
174		iunss 4753	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑦 ∈ 𝑣 ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ ∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐) ↔ ∀𝑦 ∈ 𝑣 ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ ∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
175	173, 174	sylibr 225	. . . . . . . . . . . . . 14 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ∪ 𝑦 ∈ 𝑣 ((𝑦(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ⊆ ∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
176	124, 175	eqsstr3d 3837	. . . . . . . . . . . . 13 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → 𝑌 ⊆ ∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐))
177	117, 176	eqssd 3815	. . . . . . . . . . . 12 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ∪ 𝑧 ∈ ran 𝑓(𝑧(ball‘𝑁)𝑐) = 𝑌)
178	106, 177	syl5eq 2852	. . . . . . . . . . 11 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑁)𝑐) = 𝑌)
179		iuneq1 4726	. . . . . . . . . . . . 13 ⊢ (𝑤 = ran 𝑓 → ∪ 𝑥 ∈ 𝑤 (𝑥(ball‘𝑁)𝑐) = ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑁)𝑐))
180	179	eqeq1d 2808	. . . . . . . . . . . 12 ⊢ (𝑤 = ran 𝑓 → (∪ 𝑥 ∈ 𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌 ↔ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑁)𝑐) = 𝑌))
181	180	rspcev 3502	. . . . . . . . . . 11 ⊢ ((ran 𝑓 ∈ (𝒫 𝑌 ∩ Fin) ∧ ∪ 𝑥 ∈ ran 𝑓(𝑥(ball‘𝑁)𝑐) = 𝑌) → ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin)∪ 𝑥 ∈ 𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌)
182	104, 178, 181	syl2anc 575	. . . . . . . . . 10 ⊢ (((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) ∧ (dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2)))))) → ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin)∪ 𝑥 ∈ 𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌)
183	182	ex 399	. . . . . . . . 9 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → ((dom 𝑓 ⊆ 𝑣 ∧ 𝑓:dom 𝑓⟶𝑌 ∧ ∀𝑥 ∈ 𝑣 (((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅ → 𝑥 ∈ (^◡𝑓 “ (𝑥(ball‘𝑀)(𝑐 / 2))))) → ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin)∪ 𝑥 ∈ 𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
184	91, 183	syld 47	. . . . . . . 8 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → ((𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))) → ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin)∪ 𝑥 ∈ 𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
185	184	exlimdv 2024	. . . . . . 7 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → (∃𝑓(𝑓:{𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅}⟶𝑌 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝑣 ∣ ((𝑥(ball‘𝑀)(𝑐 / 2)) ∩ 𝑌) ≠ ∅} (𝑓‘𝑦) ∈ (𝑦(ball‘𝑀)(𝑐 / 2))) → ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin)∪ 𝑥 ∈ 𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
186	68, 185	mpd 15	. . . . . 6 ⊢ ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) ∧ (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)))) → ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin)∪ 𝑥 ∈ 𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌)
187	186	rexlimdvaa 3220	. . . . 5 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) → (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)(𝑐 / 2)) → ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin)∪ 𝑥 ∈ 𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
188	47, 187	syld 47	. . . 4 ⊢ (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑐 ∈ ℝ⁺) → (∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) → ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin)∪ 𝑥 ∈ 𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
189	188	ralrimdva 3157	. . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) → ∀𝑐 ∈ ℝ⁺ ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin)∪ 𝑥 ∈ 𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
190		istotbnd3 33876	. . . . 5 ⊢ (𝑁 ∈ (TotBnd‘𝑌) ↔ (𝑁 ∈ (Met‘𝑌) ∧ ∀𝑐 ∈ ℝ⁺ ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin)∪ 𝑥 ∈ 𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
191	190	baib 527	. . . 4 ⊢ (𝑁 ∈ (Met‘𝑌) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑐 ∈ ℝ⁺ ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin)∪ 𝑥 ∈ 𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
192	3, 191	syl 17	. . 3 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑐 ∈ ℝ⁺ ∃𝑤 ∈ (𝒫 𝑌 ∩ Fin)∪ 𝑥 ∈ 𝑤 (𝑥(ball‘𝑁)𝑐) = 𝑌))
193	189, 192	sylibrd 250	. 2 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑) → 𝑁 ∈ (TotBnd‘𝑌)))
194	39, 193	impbid 203	1 ⊢ ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ⁺ ∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 ⊆ ∪ 𝑥 ∈ 𝑣 (𝑥(ball‘𝑀)𝑑)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 197 ∧ wa 384 ∧ w3a 1100 = wceq 1637 ∃wex 1859 ∈ wcel 2156 ≠ wne 2978 ∀wral 3096 ∃wrex 3097 {crab 3100 ∩ cin 3768 ⊆ wss 3769 ∅c0 4116 𝒫 cpw 4351 ∪ ciun 4712 × cxp 5309 ^◡ccnv 5310 dom cdm 5311 ran crn 5312 ↾ cres 5313 “ cima 5314 Fn wfn 6092 ⟶wf 6093 ‘cfv 6097 (class class class)co 6870 Fincfn 8188 ℝcr 10216 ℝ^*cxr 10354 / cdiv 10965 2c2 11352 ℝ⁺crp 12042 ∞Metcxmt 19935 Metcme 19936 ballcbl 19937 TotBndctotbnd 33871

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1877 ax-4 1894 ax-5 2001 ax-6 2068 ax-7 2104 ax-8 2158 ax-9 2165 ax-10 2185 ax-11 2201 ax-12 2214 ax-13 2420 ax-ext 2784 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5096 ax-un 7175 ax-cnex 10273 ax-resscn 10274 ax-1cn 10275 ax-icn 10276 ax-addcl 10277 ax-addrcl 10278 ax-mulcl 10279 ax-mulrcl 10280 ax-mulcom 10281 ax-addass 10282 ax-mulass 10283 ax-distr 10284 ax-i2m1 10285 ax-1ne0 10286 ax-1rid 10287 ax-rnegex 10288 ax-rrecex 10289 ax-cnre 10290 ax-pre-lttri 10291 ax-pre-lttrn 10292 ax-pre-ltadd 10293 ax-pre-mulgt0 10294

This theorem depends on definitions: df-bi 198 df-an 385 df-or 866 df-3or 1101 df-3an 1102 df-tru 1641 df-ex 1860 df-nf 1864 df-sb 2061 df-eu 2634 df-mo 2635 df-clab 2793 df-cleq 2799 df-clel 2802 df-nfc 2937 df-ne 2979 df-nel 3082 df-ral 3101 df-rex 3102 df-reu 3103 df-rmo 3104 df-rab 3105 df-v 3393 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4117 df-if 4280 df-pw 4353 df-sn 4371 df-pr 4373 df-tp 4375 df-op 4377 df-uni 4631 df-int 4670 df-iun 4714 df-br 4845 df-opab 4907 df-mpt 4924 df-tr 4947 df-id 5219 df-eprel 5224 df-po 5232 df-so 5233 df-fr 5270 df-we 5272 df-xp 5317 df-rel 5318 df-cnv 5319 df-co 5320 df-dm 5321 df-rn 5322 df-res 5323 df-ima 5324 df-pred 5893 df-ord 5939 df-on 5940 df-lim 5941 df-suc 5942 df-iota 6060 df-fun 6099 df-fn 6100 df-f 6101 df-f1 6102 df-fo 6103 df-f1o 6104 df-fv 6105 df-riota 6831 df-ov 6873 df-oprab 6874 df-mpt2 6875 df-om 7292 df-1st 7394 df-2nd 7395 df-wrecs 7638 df-recs 7700 df-rdg 7738 df-1o 7792 df-oadd 7796 df-er 7975 df-map 8090 df-en 8189 df-dom 8190 df-sdom 8191 df-fin 8192 df-pnf 10357 df-mnf 10358 df-xr 10359 df-ltxr 10360 df-le 10361 df-sub 10549 df-neg 10550 df-div 10966 df-2 11360 df-rp 12043 df-xneg 12158 df-xadd 12159 df-xmul 12160 df-psmet 19942 df-xmet 19943 df-met 19944 df-bl 19945 df-totbnd 33873

This theorem is referenced by: sstotbnd 33880 sstotbnd3 33881

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