Theorem metrest 24804

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 4181	. . . . . . . . . 10 ⊢ (𝑢 ∩ 𝑌) ⊆ 𝑢
2		metrest.3	. . . . . . . . . . . . 13 ⊢ 𝐽 = (MetOpen‘𝐶)
3	2	elmopn2 24725	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (∞Met‘𝑋) → (𝑢 ∈ 𝐽 ↔ (𝑢 ⊆ 𝑋 ∧ ∀𝑦 ∈ 𝑢 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)))
4	3	simplbda 505	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢 ∈ 𝐽) → ∀𝑦 ∈ 𝑢 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)
5	4	adantlr 728	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑢 ∈ 𝐽) → ∀𝑦 ∈ 𝑢 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)
6		ssralv 3999	. . . . . . . . . 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 69	. . . . . . . . 9 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑢 ∈ 𝐽) → ∀𝑦 ∈ (𝑢 ∩ 𝑌)∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)
8		ssrin 4186	. . . . . . . . . . 11 ⊢ ((𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌))
9	8	reximi 3100	. . . . . . . . . 10 ⊢ (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌))
10	9	ralimi 3099	. . . . . . . . 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 18	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑢 ∈ 𝐽) → ∀𝑦 ∈ (𝑢 ∩ 𝑌)∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌))
12		inss2 4182	. . . . . . . 8 ⊢ (𝑢 ∩ 𝑌) ⊆ 𝑌
13	11, 12	jctil 529	. . . . . . 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 3955	. . . . . . . 8 ⊢ (𝑥 = (𝑢 ∩ 𝑌) → (𝑥 ⊆ 𝑌 ↔ (𝑢 ∩ 𝑌) ⊆ 𝑌))
15		sseq2 3956	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 ∩ 𝑌) → (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌)))
16	15	rexbidv 3186	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 ∩ 𝑌) → (∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌)))
17	16	raleqbi1dv 3329	. . . . . . . 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 644	. . . . . . 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 250	. . . . . 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 3163	. . . . 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 24720	. . . . . . . . 9 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
22	21	ad2antrr 739	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → 𝐽 ∈ Top)
23		ssel2 3925	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ⊆ 𝑌 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑌)
24		ssel2 3925	. . . . . . . . . . . . . . . 16 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
25		rpxr 13099	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
26	2	blopn 24780	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐶)𝑟) ∈ 𝐽)
27		eleq1a 2855	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦(ball‘𝐶)𝑟) ∈ 𝐽 → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
28	26, 27	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
29	28	3expa 1136	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ*) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
30	25, 29	sylan2 605	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
31	30	rexlimdva 3163	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
32	24, 31	sylan2 605	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
33	32	anassrs 473	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑦 ∈ 𝑌) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
34	23, 33	sylan2 605	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ 𝑦 ∈ 𝑥)) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
35	34	anassrs 473	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
36	35	rexlimdva 3163	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑌) → (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
37	36	adantrd 497	. . . . . . . . . 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 730	. . . . . . . . 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 4014	. . . . . . . 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 23176	. . . . . . . 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 596	. . . . . . 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 7415	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑢 → (𝑦(ball‘𝐶)𝑟) = (𝑢(ball‘𝐶)𝑟))
43	42	ineq1d 4164	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑢 → ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) = ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌))
44	43	sseq1d 3961	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
45	44	rexbidv 3186	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑢 → (∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
46	45	rspccv 3573	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → (𝑢 ∈ 𝑥 → ∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
47	46	ad2antll 742	. . . . . . . . . . . . 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 3924	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ 𝑌 → (𝑢 ∈ 𝑥 → 𝑢 ∈ 𝑌))
49		ssel 3924	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 ⊆ 𝑋 → (𝑢 ∈ 𝑌 → 𝑢 ∈ 𝑋))
50		blcntr 24693	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → 𝑢 ∈ (𝑢(ball‘𝐶)𝑟))
51	50	a1d 26	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢 ∈ 𝑋 ∧ 𝑟 ∈ ℝ+) → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → 𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))
52	51	ancld 560	. . . . . . . . . . . . . . . . . . 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 1136	. . . . . . . . . . . . . . . . . 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 3175	. . . . . . . . . . . . . . . . 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 418	. . . . . . . . . . . . . . . 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 518	. . . . . . . . . . . . . . 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 518	. . . . . . . . . . . . . 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 730	. . . . . . . . . . . . 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 44	. . . . . . . . . . . 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 2846	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ↔ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))
61	44, 60	anbi12d 644	. . . . . . . . . . . . . . 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 3186	. . . . . . . . . . . . . 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 3576	. . . . . . . . . . . . 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 418	. . . . . . . . . . . 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 31	. . . . . . . . . . 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 783	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → 𝑥 ⊆ 𝑌)
67	66	sseld 3929	. . . . . . . . . . 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 522	. . . . . . . . . 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 3914	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ↔ (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ∧ 𝑢 ∈ 𝑌))
70		ssel2 3925	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌)) → 𝑢 ∈ 𝑥)
71	69, 70	sylan2br 607	. . . . . . . . . . . . . 14 ⊢ ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ∧ 𝑢 ∈ 𝑌)) → 𝑢 ∈ 𝑥)
72	71	expr 462	. . . . . . . . . . . . 13 ⊢ ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢 ∈ 𝑌 → 𝑢 ∈ 𝑥))
73	72	rexlimivw 3159	. . . . . . . . . . . 12 ⊢ (∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢 ∈ 𝑌 → 𝑢 ∈ 𝑥))
74	73	rexlimivw 3159	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢 ∈ 𝑌 → 𝑢 ∈ 𝑥))
75	74	imp 412	. . . . . . . . . 10 ⊢ ((∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ∧ 𝑢 ∈ 𝑌) → 𝑢 ∈ 𝑥)
76	68, 75	impbid1 228	. . . . . . . . 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 3914	. . . . . . . . . 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 4880	. . . . . . . . . . . 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 466	. . . . . . . . . . . . . 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 474	. . . . . . . . . . . . . 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 3192	. . . . . . . . . . . . . . . 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 3109	. . . . . . . . . . . . . . 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 3192	. . . . . . . . . . . . . . 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 279	. . . . . . . . . . . . . 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 300	. . . . . . . . . . . . 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 1881	. . . . . . . . . . . 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 7441	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦(ball‘𝐶)𝑟) ∈ V
88		ineq1 4158	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑦(ball‘𝐶)𝑟) → (𝑧 ∩ 𝑌) = ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌))
89	88	sseq1d 3961	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑦(ball‘𝐶)𝑟) → ((𝑧 ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
90		eleq2 2849	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑦(ball‘𝐶)𝑟) → (𝑢 ∈ 𝑧 ↔ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
91	89, 90	anbi12d 644	. . . . . . . . . . . . . . . . 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 3498	. . . . . . . . . . . . . . . 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 3109	. . . . . . . . . . . . . . 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 3289	. . . . . . . . . . . . . . 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 280	. . . . . . . . . . . . . 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 3109	. . . . . . . . . . . . 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 3289	. . . . . . . . . . . . 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 279	. . . . . . . . . . . 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 300	. . . . . . . . . . 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 636	. . . . . . . . . 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 279	. . . . . . . . 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	bitrdi 290	. . . . . . . 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 2758	. . . . . . 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 4158	. . . . . . . 8 ⊢ (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	rspceeqv 3598	. . . . . . 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))
106	41, 103, 105	syl2anc 596	. . . . . 6 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → ∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌))
107	106	ex 418	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → ((𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥) → ∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌)))
108	20, 107	impbid 215	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌) ↔ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
109		simpr 490	. . . . . . . . . . 11 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
110	24, 109	elind 4145	. . . . . . . . . 10 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ (𝑋 ∩ 𝑌))
111		metrest.1	. . . . . . . . . . . . . . 15 ⊢ 𝐷 = (𝐶 ↾ (𝑌 × 𝑌))
112	111	blres 24711	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) = ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌))
113	112	sseq1d 3961	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌) ∧ 𝑟 ∈ ℝ*) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
114	113	3expa 1136	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌)) ∧ 𝑟 ∈ ℝ*) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
115	25, 114	sylan2 605	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌)) ∧ 𝑟 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
116	115	rexbidva 3184	. . . . . . . . . 10 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌)) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
117	110, 116	sylan2 605	. . . . . . . . 9 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
118	117	anassrs 473	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑦 ∈ 𝑌) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
119	23, 118	sylan2 605	. . . . . . 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))
120	119	anassrs 473	. . . . . 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))
121	120	ralbidva 3183	. . . . 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))
122	121	pm5.32da 590	. . . 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)))
123	108, 122	bitr4d 285	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌) ↔ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)))
124		id 23	. . . . 5 ⊢ (𝑌 ⊆ 𝑋 → 𝑌 ⊆ 𝑋)
125	2	mopnm 24724	. . . . 5 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝑋 ∈ 𝐽)
126		ssexg 5280	. . . . 5 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑌 ∈ V)
127	124, 125, 126	syl2anr 609	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
128		elrest 17559	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝑥 ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌)))
129	21, 127, 128	syl2an2r 698	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌)))
130		xmetres2 24641	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐶 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌))
131	111, 130	eqeltrid 2864	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → 𝐷 ∈ (∞Met‘𝑌))
132		metrest.4	. . . . 5 ⊢ 𝐾 = (MetOpen‘𝐷)
133	132	elmopn2 24725	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑌) → (𝑥 ∈ 𝐾 ↔ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)))
134	131, 133	syl 18	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∈ 𝐾 ↔ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)))
135	123, 129, 134	3bitr4d 314	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∈ (𝐽 ↾t 𝑌) ↔ 𝑥 ∈ 𝐾))
136	135	eqrdv 2758	1 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) = 𝐾)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450   ∩ cin 3897   ⊆ wss 3898  ∪ cuni 4866   × cxp 5645   ↾ cres 5649  ‘cfv 6527  (class class class)co 7408  ℝ^*cxr 11313  ℝ⁺crp 13089   ↾_t crest 17552  ∞Metcxmet 21624  ballcbl 21626  MetOpencmopn 21629  Topctop 23172

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734  ax-cnex 11227  ax-resscn 11228  ax-1cn 11229  ax-icn 11230  ax-addcl 11231  ax-addrcl 11232  ax-mulcl 11233  ax-mulrcl 11234  ax-mulcom 11235  ax-addass 11236  ax-mulass 11237  ax-distr 11238  ax-i2m1 11239  ax-1ne0 11240  ax-1rid 11241  ax-rnegex 11242  ax-rrecex 11243  ax-cnre 11244  ax-pre-lttri 11245  ax-pre-lttrn 11246  ax-pre-ltadd 11247  ax-pre-mulgt0 11248  ax-pre-sup 11249

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-nel 3062  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-lim 6356  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-rdg 8396  df-er 8695  df-map 8827  df-en 8952  df-dom 8953  df-sdom 8954  df-sup 9412  df-inf 9413  df-pnf 11316  df-mnf 11317  df-xr 11318  df-ltxr 11319  df-le 11320  df-sub 11514  df-neg 11515  df-div 11943  df-nn 12305  df-2 12374  df-n0 12576  df-z 12663  df-uz 12935  df-q 13045  df-rp 13090  df-xneg 13210  df-xadd 13211  df-xmul 13212  df-rest 17554  df-topgen 17575  df-psmet 21631  df-xmet 21632  df-bl 21634  df-mopn 21635  df-top 23173  df-topon 23190  df-bases 23225

This theorem is used by:  ressxms  24805  nrginvrcn  24972  resubmet  25082  tgioo2  25083  metdscn2  25138  divcn  25150  dfii3  25165  cncfcn  25192  metsscmetcld  25597  cmetss  25598  minveclem4a  25712  ftc1lem6  26322  ulmdvlem3  26692  abelth  26731  cxpcn3  27039  rlimcnp  27256  minvecolem4b  31413  minvecolem4  31415  hhsscms  31813  ftc1cnnc  38530