Theorem metrest 22548

Description: Two alternate formulations of a subspace topology of a metric space topology. (Contributed by Jeff Hankins, 19-Aug-2009.) (Proof shortened by Mario Carneiro, 5-Jan-2014.)

Hypotheses

Ref	Expression
metrest.1	⊢ 𝐷 = (𝐶 ↾ (𝑌 × 𝑌))
metrest.3	⊢ 𝐽 = (MetOpen‘𝐶)
metrest.4	⊢ 𝐾 = (MetOpen‘𝐷)

Assertion

Ref	Expression
metrest	⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) = 𝐾)

Proof of Theorem metrest

Dummy variables 𝑢 𝑟 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		inss1 3981	. . . . . . . . . 10 ⊢ (𝑢 ∩ 𝑌) ⊆ 𝑢
2		metrest.3	. . . . . . . . . . . . 13 ⊢ 𝐽 = (MetOpen‘𝐶)
3	2	elmopn2 22469	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (∞Met‘𝑋) → (𝑢 ∈ 𝐽 ↔ (𝑢 ⊆ 𝑋 ∧ ∀𝑦 ∈ 𝑢 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)))
4	3	simplbda 487	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢 ∈ 𝐽) → ∀𝑦 ∈ 𝑢 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)
5	4	adantlr 694	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑢 ∈ 𝐽) → ∀𝑦 ∈ 𝑢 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)
6		ssralv 3815	. . . . . . . . . 10 ⊢ ((𝑢 ∩ 𝑌) ⊆ 𝑢 → (∀𝑦 ∈ 𝑢 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ∀𝑦 ∈ (𝑢 ∩ 𝑌)∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢))
7	1, 5, 6	mpsyl 68	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑢 ∈ 𝐽) → ∀𝑦 ∈ (𝑢 ∩ 𝑌)∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)
8		ssrin 3986	. . . . . . . . . . 11 ⊢ ((𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌))
9	8	reximi 3159	. . . . . . . . . 10 ⊢ (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌))
10	9	ralimi 3101	. . . . . . . . 9 ⊢ (∀𝑦 ∈ (𝑢 ∩ 𝑌)∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ∀𝑦 ∈ (𝑢 ∩ 𝑌)∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌))
11	7, 10	syl 17	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑢 ∈ 𝐽) → ∀𝑦 ∈ (𝑢 ∩ 𝑌)∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌))
12		inss2 3982	. . . . . . . 8 ⊢ (𝑢 ∩ 𝑌) ⊆ 𝑌
13	11, 12	jctil 509	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑢 ∈ 𝐽) → ((𝑢 ∩ 𝑌) ⊆ 𝑌 ∧ ∀𝑦 ∈ (𝑢 ∩ 𝑌)∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌)))
14		sseq1 3775	. . . . . . . 8 ⊢ (𝑥 = (𝑢 ∩ 𝑌) → (𝑥 ⊆ 𝑌 ↔ (𝑢 ∩ 𝑌) ⊆ 𝑌))
15		sseq2 3776	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 ∩ 𝑌) → (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌)))
16	15	rexbidv 3200	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 ∩ 𝑌) → (∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌)))
17	16	raleqbi1dv 3295	. . . . . . . 8 ⊢ (𝑥 = (𝑢 ∩ 𝑌) → (∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∀𝑦 ∈ (𝑢 ∩ 𝑌)∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌)))
18	14, 17	anbi12d 616	. . . . . . 7 ⊢ (𝑥 = (𝑢 ∩ 𝑌) → ((𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥) ↔ ((𝑢 ∩ 𝑌) ⊆ 𝑌 ∧ ∀𝑦 ∈ (𝑢 ∩ 𝑌)∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌))))
19	13, 18	syl5ibrcom 237	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑢 ∈ 𝐽) → (𝑥 = (𝑢 ∩ 𝑌) → (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
20	19	rexlimdva 3179	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌) → (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
21	2	mopntop 22464	. . . . . . . . 9 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
22	21	ad2antrr 705	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → 𝐽 ∈ Top)
23		ssel2 3747	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ⊆ 𝑌 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑌)
24		ssel2 3747	. . . . . . . . . . . . . . . 16 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
25		rpxr 12042	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
26	2	blopn 22524	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐶)𝑟) ∈ 𝐽)
27		eleq1a 2845	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦(ball‘𝐶)𝑟) ∈ 𝐽 → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
28	26, 27	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
29	28	3expa 1111	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ*) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
30	25, 29	sylan2 580	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
31	30	rexlimdva 3179	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
32	24, 31	sylan2 580	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
33	32	anassrs 453	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑦 ∈ 𝑌) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
34	23, 33	sylan2 580	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ 𝑦 ∈ 𝑥)) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
35	34	anassrs 453	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
36	35	rexlimdva 3179	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑌) → (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
37	36	adantrd 479	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑌) → ((∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥) → 𝑧 ∈ 𝐽))
38	37	adantrr 696	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → ((∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥) → 𝑧 ∈ 𝐽))
39	38	abssdv 3825	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} ⊆ 𝐽)
40		uniopn 20921	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} ⊆ 𝐽) → ∪ {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} ∈ 𝐽)
41	22, 39, 40	syl2anc 573	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → ∪ {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} ∈ 𝐽)
42		oveq1 6802	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑢 → (𝑦(ball‘𝐶)𝑟) = (𝑢(ball‘𝐶)𝑟))
43	42	ineq1d 3964	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑢 → ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) = ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌))
44	43	sseq1d 3781	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
45	44	rexbidv 3200	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑢 → (∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
46	45	rspccv 3457	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → (𝑢 ∈ 𝑥 → ∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
47	46	ad2antll 708	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢 ∈ 𝑥 → ∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
48		ssel 3746	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ 𝑌 → (𝑢 ∈ 𝑥 → 𝑢 ∈ 𝑌))
49		ssel 3746	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 ⊆ 𝑋 → (𝑢 ∈ 𝑌 → 𝑢 ∈ 𝑋))
50		blcntr 22437	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → 𝑢 ∈ (𝑢(ball‘𝐶)𝑟))
51	50	a1d 25	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → 𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))
52	51	ancld 540	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
53	52	3expa 1111	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
54	53	reximdva 3165	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢 ∈ 𝑋) → (∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → ∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
55	54	ex 397	. . . . . . . . . . . . . . . 16 ⊢ (𝐶 ∈ (∞Met‘𝑋) → (𝑢 ∈ 𝑋 → (∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → ∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))))
56	49, 55	sylan9r 498	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑢 ∈ 𝑌 → (∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → ∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))))
57	48, 56	sylan9r 498	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑌) → (𝑢 ∈ 𝑥 → (∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → ∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))))
58	57	adantrr 696	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢 ∈ 𝑥 → (∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → ∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))))
59	47, 58	mpdd 43	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢 ∈ 𝑥 → ∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
60	42	eleq2d 2836	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ↔ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))
61	44, 60	anbi12d 616	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑢 → ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ↔ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
62	61	rexbidv 3200	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑢 → (∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ↔ ∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
63	62	rspcev 3460	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑥 ∧ ∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟))) → ∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
64	63	ex 397	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ 𝑥 → (∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟)) → ∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟))))
65	59, 64	sylcom 30	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢 ∈ 𝑥 → ∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟))))
66		simprl 754	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → 𝑥 ⊆ 𝑌)
67	66	sseld 3751	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢 ∈ 𝑥 → 𝑢 ∈ 𝑌))
68	65, 67	jcad 502	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢 ∈ 𝑥 → (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ∧ 𝑢 ∈ 𝑌)))
69		elin 3947	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ↔ (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ∧ 𝑢 ∈ 𝑌))
70		ssel2 3747	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌)) → 𝑢 ∈ 𝑥)
71	69, 70	sylan2br 582	. . . . . . . . . . . . . 14 ⊢ ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ∧ 𝑢 ∈ 𝑌)) → 𝑢 ∈ 𝑥)
72	71	expr 444	. . . . . . . . . . . . 13 ⊢ ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢 ∈ 𝑌 → 𝑢 ∈ 𝑥))
73	72	rexlimivw 3177	. . . . . . . . . . . 12 ⊢ (∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢 ∈ 𝑌 → 𝑢 ∈ 𝑥))
74	73	rexlimivw 3177	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢 ∈ 𝑌 → 𝑢 ∈ 𝑥))
75	74	imp 393	. . . . . . . . . 10 ⊢ ((∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ∧ 𝑢 ∈ 𝑌) → 𝑢 ∈ 𝑥)
76	68, 75	impbid1 215	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢 ∈ 𝑥 ↔ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ∧ 𝑢 ∈ 𝑌)))
77		elin 3947	. . . . . . . . . 10 ⊢ (𝑢 ∈ (∪ {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} ∩ 𝑌) ↔ (𝑢 ∈ ∪ {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} ∧ 𝑢 ∈ 𝑌))
78		eluniab 4586	. . . . . . . . . . . 12 ⊢ (𝑢 ∈ ∪ {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} ↔ ∃𝑧(𝑢 ∈ 𝑧 ∧ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)))
79		ancom 448	. . . . . . . . . . . . . 14 ⊢ ((𝑢 ∈ 𝑧 ∧ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)) ↔ ((∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥) ∧ 𝑢 ∈ 𝑧))
80		anass 454	. . . . . . . . . . . . . 14 ⊢ (((∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥) ∧ 𝑢 ∈ 𝑧) ↔ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)))
81		r19.41v 3237	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)) ↔ (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)))
82	81	rexbii 3189	. . . . . . . . . . . . . . 15 ⊢ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)) ↔ ∃𝑦 ∈ 𝑥 (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)))
83		r19.41v 3237	. . . . . . . . . . . . . . 15 ⊢ (∃𝑦 ∈ 𝑥 (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)) ↔ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)))
84	82, 83	bitr2i 265	. . . . . . . . . . . . . 14 ⊢ ((∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)) ↔ ∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)))
85	79, 80, 84	3bitri 286	. . . . . . . . . . . . 13 ⊢ ((𝑢 ∈ 𝑧 ∧ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)) ↔ ∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)))
86	85	exbii 1924	. . . . . . . . . . . 12 ⊢ (∃𝑧(𝑢 ∈ 𝑧 ∧ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)) ↔ ∃𝑧∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)))
87		ovex 6826	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦(ball‘𝐶)𝑟) ∈ V
88		ineq1 3958	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑦(ball‘𝐶)𝑟) → (𝑧 ∩ 𝑌) = ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌))
89	88	sseq1d 3781	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑦(ball‘𝐶)𝑟) → ((𝑧 ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
90		eleq2 2839	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑦(ball‘𝐶)𝑟) → (𝑢 ∈ 𝑧 ↔ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
91	89, 90	anbi12d 616	. . . . . . . . . . . . . . . . 17 ⊢ (𝑧 = (𝑦(ball‘𝐶)𝑟) → (((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧) ↔ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟))))
92	87, 91	ceqsexv 3394	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑧(𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)) ↔ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
93	92	rexbii 3189	. . . . . . . . . . . . . . 15 ⊢ (∃𝑟 ∈ ℝ+ ∃𝑧(𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)) ↔ ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
94		rexcom4 3377	. . . . . . . . . . . . . . 15 ⊢ (∃𝑟 ∈ ℝ+ ∃𝑧(𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)) ↔ ∃𝑧∃𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)))
95	93, 94	bitr3i 266	. . . . . . . . . . . . . 14 ⊢ (∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ↔ ∃𝑧∃𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)))
96	95	rexbii 3189	. . . . . . . . . . . . 13 ⊢ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ↔ ∃𝑦 ∈ 𝑥 ∃𝑧∃𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)))
97		rexcom4 3377	. . . . . . . . . . . . 13 ⊢ (∃𝑦 ∈ 𝑥 ∃𝑧∃𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)) ↔ ∃𝑧∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)))
98	96, 97	bitr2i 265	. . . . . . . . . . . 12 ⊢ (∃𝑧∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧 ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ 𝑧)) ↔ ∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
99	78, 86, 98	3bitri 286	. . . . . . . . . . 11 ⊢ (𝑢 ∈ ∪ {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} ↔ ∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
100	99	anbi1i 610	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ∪ {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} ∧ 𝑢 ∈ 𝑌) ↔ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ∧ 𝑢 ∈ 𝑌))
101	77, 100	bitr2i 265	. . . . . . . . 9 ⊢ ((∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ∧ 𝑢 ∈ 𝑌) ↔ 𝑢 ∈ (∪ {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} ∩ 𝑌))
102	76, 101	syl6bb 276	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢 ∈ 𝑥 ↔ 𝑢 ∈ (∪ {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} ∩ 𝑌)))
103	102	eqrdv 2769	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → 𝑥 = (∪ {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} ∩ 𝑌))
104		ineq1 3958	. . . . . . . . 9 ⊢ (𝑢 = ∪ {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} → (𝑢 ∩ 𝑌) = (∪ {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} ∩ 𝑌))
105	104	eqeq2d 2781	. . . . . . . 8 ⊢ (𝑢 = ∪ {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} → (𝑥 = (𝑢 ∩ 𝑌) ↔ 𝑥 = (∪ {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} ∩ 𝑌)))
106	105	rspcev 3460	. . . . . . 7 ⊢ ((∪ {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} ∈ 𝐽 ∧ 𝑥 = (∪ {𝑧 ∣ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧 ∩ 𝑌) ⊆ 𝑥)} ∩ 𝑌)) → ∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌))
107	41, 103, 106	syl2anc 573	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → ∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌))
108	107	ex 397	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → ((𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥) → ∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌)))
109	20, 108	impbid 202	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌) ↔ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
110		simpr 471	. . . . . . . . . . 11 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
111	24, 110	elind 3949	. . . . . . . . . 10 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ (𝑋 ∩ 𝑌))
112		metrest.1	. . . . . . . . . . . . . . 15 ⊢ 𝐷 = (𝐶 ↾ (𝑌 × 𝑌))
113	112	blres 22455	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) = ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌))
114	113	sseq1d 3781	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌) ∧ 𝑟 ∈ ℝ*) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
115	114	3expa 1111	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌)) ∧ 𝑟 ∈ ℝ*) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
116	25, 115	sylan2 580	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌)) ∧ 𝑟 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
117	116	rexbidva 3197	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌)) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
118	111, 117	sylan2 580	. . . . . . . . 9 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
119	118	anassrs 453	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑦 ∈ 𝑌) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
120	23, 119	sylan2 580	. . . . . . 7 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ 𝑦 ∈ 𝑥)) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
121	120	anassrs 453	. . . . . 6 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
122	121	ralbidva 3134	. . . . 5 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑌) → (∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
123	122	pm5.32da 568	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → ((𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥) ↔ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
124	109, 123	bitr4d 271	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌) ↔ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)))
125	21	adantr 466	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → 𝐽 ∈ Top)
126		id 22	. . . . 5 ⊢ (𝑌 ⊆ 𝑋 → 𝑌 ⊆ 𝑋)
127	2	mopnm 22468	. . . . 5 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝑋 ∈ 𝐽)
128		ssexg 4939	. . . . 5 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑌 ∈ V)
129	126, 127, 128	syl2anr 584	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
130		elrest 16295	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝑥 ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌)))
131	125, 129, 130	syl2anc 573	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌)))
132		xmetres2 22385	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐶 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌))
133	112, 132	syl5eqel 2854	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → 𝐷 ∈ (∞Met‘𝑌))
134		metrest.4	. . . . 5 ⊢ 𝐾 = (MetOpen‘𝐷)
135	134	elmopn2 22469	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑌) → (𝑥 ∈ 𝐾 ↔ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)))
136	133, 135	syl 17	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∈ 𝐾 ↔ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)))
137	124, 131, 136	3bitr4d 300	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∈ (𝐽 ↾t 𝑌) ↔ 𝑥 ∈ 𝐾))
138	137	eqrdv 2769	1 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) = 𝐾)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1071   = wceq 1631  ∃wex 1852   ∈ wcel 2145  {cab 2757  ∀wral 3061  ∃wrex 3062  Vcvv 3351   ∩ cin 3722   ⊆ wss 3723  ∪ cuni 4575   × cxp 5248   ↾ cres 5252  ‘cfv 6030  (class class class)co 6795  ℝ^*cxr 10278  ℝ⁺crp 12034   ↾_t crest 16288  ∞Metcxmt 19945  ballcbl 19947  MetOpencmopn 19950  Topctop 20917

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-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7099  ax-cnex 10197  ax-resscn 10198  ax-1cn 10199  ax-icn 10200  ax-addcl 10201  ax-addrcl 10202  ax-mulcl 10203  ax-mulrcl 10204  ax-mulcom 10205  ax-addass 10206  ax-mulass 10207  ax-distr 10208  ax-i2m1 10209  ax-1ne0 10210  ax-1rid 10211  ax-rnegex 10212  ax-rrecex 10213  ax-cnre 10214  ax-pre-lttri 10215  ax-pre-lttrn 10216  ax-pre-ltadd 10217  ax-pre-mulgt0 10218  ax-pre-sup 10219

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 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-riota 6756  df-ov 6798  df-oprab 6799  df-mpt2 6800  df-om 7216  df-1st 7318  df-2nd 7319  df-wrecs 7562  df-recs 7624  df-rdg 7662  df-er 7899  df-map 8014  df-en 8113  df-dom 8114  df-sdom 8115  df-sup 8507  df-inf 8508  df-pnf 10281  df-mnf 10282  df-xr 10283  df-ltxr 10284  df-le 10285  df-sub 10473  df-neg 10474  df-div 10890  df-nn 11226  df-2 11284  df-n0 11499  df-z 11584  df-uz 11893  df-q 11996  df-rp 12035  df-xneg 12150  df-xadd 12151  df-xmul 12152  df-rest 16290  df-topgen 16311  df-psmet 19952  df-xmet 19953  df-bl 19955  df-mopn 19956  df-top 20918  df-topon 20935  df-bases 20970

This theorem is referenced by:  ressxms  22549  nrginvrcn  22715  resubmet  22824  tgioo2  22825  metdscn2  22879  divcn  22890  dfii3  22905  cncfcn  22931  cmetss  23331  minveclem4a  23419  ftc1lem6  24023  ulmdvlem3  24375  abelth  24414  cxpcn3  24709  rlimcnp  24912  minvecolem4b  28073  minvecolem4  28075  hhsscms  28475  ftc1cnnc  33815