Theorem metrest 22329

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 3833	. . . . . . . . . 10 ⊢ (𝑢 ∩ 𝑌) ⊆ 𝑢
2		metrest.3	. . . . . . . . . . . . 13 ⊢ 𝐽 = (MetOpen‘𝐶)
3	2	elmopn2 22250	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (∞Met‘𝑋) → (𝑢 ∈ 𝐽 ↔ (𝑢 ⊆ 𝑋 ∧ ∀𝑦 ∈ 𝑢 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)))
4	3	simplbda 654	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢 ∈ 𝐽) → ∀𝑦 ∈ 𝑢 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)
5	4	adantlr 751	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑢 ∈ 𝐽) → ∀𝑦 ∈ 𝑢 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)
6		ssralv 3666	. . . . . . . . . 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 3838	. . . . . . . . . . 11 ⊢ ((𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌))
9	8	reximi 3011	. . . . . . . . . 10 ⊢ (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌))
10	9	ralimi 2952	. . . . . . . . 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 3834	. . . . . . . 8 ⊢ (𝑢 ∩ 𝑌) ⊆ 𝑌
13	11, 12	jctil 560	. . . . . . 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 3626	. . . . . . . 8 ⊢ (𝑥 = (𝑢 ∩ 𝑌) → (𝑥 ⊆ 𝑌 ↔ (𝑢 ∩ 𝑌) ⊆ 𝑌))
15		sseq2 3627	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 ∩ 𝑌) → (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌)))
16	15	rexbidv 3052	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 ∩ 𝑌) → (∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌)))
17	16	raleqbi1dv 3146	. . . . . . . 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 747	. . . . . . 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 3031	. . . . 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 22245	. . . . . . . . 9 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
22	21	ad2antrr 762	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → 𝐽 ∈ Top)
23		ssel2 3598	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ⊆ 𝑌 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑌)
24		ssel2 3598	. . . . . . . . . . . . . . . 16 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
25		rpxr 11840	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
26	2	blopn 22305	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐶)𝑟) ∈ 𝐽)
27		eleq1a 2696	. . . . . . . . . . . . . . . . . . . 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 1265	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ*) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
30	25, 29	sylan2 491	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
31	30	rexlimdva 3031	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
32	24, 31	sylan2 491	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
33	32	anassrs 680	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑦 ∈ 𝑌) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
34	23, 33	sylan2 491	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ 𝑦 ∈ 𝑥)) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
35	34	anassrs 680	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
36	35	rexlimdva 3031	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑌) → (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
37	36	adantrd 484	. . . . . . . . . 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 753	. . . . . . . . 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 3676	. . . . . . . 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 20702	. . . . . . . 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 693	. . . . . . 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 6657	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑢 → (𝑦(ball‘𝐶)𝑟) = (𝑢(ball‘𝐶)𝑟))
43	42	ineq1d 3813	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑢 → ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) = ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌))
44	43	sseq1d 3632	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
45	44	rexbidv 3052	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑢 → (∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
46	45	rspccv 3306	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → (𝑢 ∈ 𝑥 → ∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
47	46	ad2antll 765	. . . . . . . . . . . . 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 3597	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ 𝑌 → (𝑢 ∈ 𝑥 → 𝑢 ∈ 𝑌))
49		ssel 3597	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 ⊆ 𝑋 → (𝑢 ∈ 𝑌 → 𝑢 ∈ 𝑋))
50		blcntr 22218	. . . . . . . . . . . . . . . . . . . . 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 576	. . . . . . . . . . . . . . . . . . 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 1265	. . . . . . . . . . . . . . . . . 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 3017	. . . . . . . . . . . . . . . . 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 450	. . . . . . . . . . . . . . . 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 690	. . . . . . . . . . . . . . 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 690	. . . . . . . . . . . . . 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 753	. . . . . . . . . . . . 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 2687	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ↔ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))
61	44, 60	anbi12d 747	. . . . . . . . . . . . . . 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 3052	. . . . . . . . . . . . . 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 3309	. . . . . . . . . . . . 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 450	. . . . . . . . . . . 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 794	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → 𝑥 ⊆ 𝑌)
67	66	sseld 3602	. . . . . . . . . . 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 555	. . . . . . . . . 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 3796	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ↔ (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ∧ 𝑢 ∈ 𝑌))
70		ssel2 3598	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌)) → 𝑢 ∈ 𝑥)
71	69, 70	sylan2br 493	. . . . . . . . . . . . . 14 ⊢ ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ∧ 𝑢 ∈ 𝑌)) → 𝑢 ∈ 𝑥)
72	71	expr 643	. . . . . . . . . . . . 13 ⊢ ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢 ∈ 𝑌 → 𝑢 ∈ 𝑥))
73	72	rexlimivw 3029	. . . . . . . . . . . 12 ⊢ (∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢 ∈ 𝑌 → 𝑢 ∈ 𝑥))
74	73	rexlimivw 3029	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢 ∈ 𝑌 → 𝑢 ∈ 𝑥))
75	74	imp 445	. . . . . . . . . 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 3796	. . . . . . . . . 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 4447	. . . . . . . . . . . 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 681	. . . . . . . . . . . . . 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 3089	. . . . . . . . . . . . . . . 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 3041	. . . . . . . . . . . . . . 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 3089	. . . . . . . . . . . . . . 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 1774	. . . . . . . . . . . 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 6678	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦(ball‘𝐶)𝑟) ∈ V
88		ineq1 3807	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑦(ball‘𝐶)𝑟) → (𝑧 ∩ 𝑌) = ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌))
89	88	sseq1d 3632	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑦(ball‘𝐶)𝑟) → ((𝑧 ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
90		eleq2 2690	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑦(ball‘𝐶)𝑟) → (𝑢 ∈ 𝑧 ↔ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
91	89, 90	anbi12d 747	. . . . . . . . . . . . . . . . 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 3242	. . . . . . . . . . . . . . . 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 3041	. . . . . . . . . . . . . . 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 3225	. . . . . . . . . . . . . . 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 3041	. . . . . . . . . . . . 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 3225	. . . . . . . . . . . . 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 731	. . . . . . . . . 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 2620	. . . . . . 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 3807	. . . . . . . . 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 2632	. . . . . . . 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 3309	. . . . . . 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 693	. . . . . 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 450	. . . . 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 477	. . . . . . . . . . 11 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
111	24, 110	elind 3798	. . . . . . . . . 10 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ (𝑋 ∩ 𝑌))
112		metrest.1	. . . . . . . . . . . . . . 15 ⊢ 𝐷 = (𝐶 ↾ (𝑌 × 𝑌))
113	112	blres 22236	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) = ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌))
114	113	sseq1d 3632	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌) ∧ 𝑟 ∈ ℝ*) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
115	114	3expa 1265	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌)) ∧ 𝑟 ∈ ℝ*) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
116	25, 115	sylan2 491	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌)) ∧ 𝑟 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
117	116	rexbidva 3049	. . . . . . . . . 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 491	. . . . . . . . 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 680	. . . . . . . 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 491	. . . . . . 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 680	. . . . . 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 2985	. . . . 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 673	. . . 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 481	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → 𝐽 ∈ Top)
126		id 22	. . . . 5 ⊢ (𝑌 ⊆ 𝑋 → 𝑌 ⊆ 𝑋)
127	2	mopnm 22249	. . . . 5 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝑋 ∈ 𝐽)
128		ssexg 4804	. . . . 5 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑌 ∈ V)
129	126, 127, 128	syl2anr 495	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
130		elrest 16088	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝑥 ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌)))
131	125, 129, 130	syl2anc 693	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌)))
132		xmetres2 22166	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐶 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌))
133	112, 132	syl5eqel 2705	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → 𝐷 ∈ (∞Met‘𝑌))
134		metrest.4	. . . . 5 ⊢ 𝐾 = (MetOpen‘𝐷)
135	134	elmopn2 22250	. . . 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 2620	1 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) = 𝐾)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∩ cin 3573   ⊆ wss 3574  ∪ cuni 4436   × cxp 5112   ↾ cres 5116  ‘cfv 5888  (class class class)co 6650  ℝ^*cxr 10073  ℝ⁺crp 11832   ↾_t crest 16081  ∞Metcxmt 19731  ballcbl 19733  MetOpencmopn 19736  Topctop 20698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-sup 8348  df-inf 8349  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-rest 16083  df-topgen 16104  df-psmet 19738  df-xmet 19739  df-bl 19741  df-mopn 19742  df-top 20699  df-topon 20716  df-bases 20750

This theorem is referenced by:  ressxms  22330  nrginvrcn  22496  resubmet  22605  tgioo2  22606  metdscn2  22660  divcn  22671  dfii3  22686  cncfcn  22712  cmetss  23113  minveclem4a  23201  ftc1lem6  23804  ulmdvlem3  24156  abelth  24195  cxpcn3  24489  rlimcnp  24692  minvecolem4b  27734  minvecolem4  27736  hhsscms  28136  ftc1cnnc  33484