Theorem metrest 24419

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 4203	. . . . . . . . . 10 ⊢ (𝑢 ∩ 𝑌) ⊆ 𝑢
2		metrest.3	. . . . . . . . . . . . 13 ⊢ 𝐽 = (MetOpen‘𝐶)
3	2	elmopn2 24340	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ (∞Met‘𝑋) → (𝑢 ∈ 𝐽 ↔ (𝑢 ⊆ 𝑋 ∧ ∀𝑦 ∈ 𝑢 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)))
4	3	simplbda 499	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢 ∈ 𝐽) → ∀𝑦 ∈ 𝑢 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)
5	4	adantlr 715	. . . . . . . . . 10 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑢 ∈ 𝐽) → ∀𝑦 ∈ 𝑢 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)
6		ssralv 4018	. . . . . . . . . 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 4208	. . . . . . . . . . 11 ⊢ ((𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌))
9	8	reximi 3068	. . . . . . . . . 10 ⊢ (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌))
10	9	ralimi 3067	. . . . . . . . 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 4204	. . . . . . . 8 ⊢ (𝑢 ∩ 𝑌) ⊆ 𝑌
13	11, 12	jctil 519	. . . . . . 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 3975	. . . . . . . 8 ⊢ (𝑥 = (𝑢 ∩ 𝑌) → (𝑥 ⊆ 𝑌 ↔ (𝑢 ∩ 𝑌) ⊆ 𝑌))
15		sseq2 3976	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 ∩ 𝑌) → (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌)))
16	15	rexbidv 3158	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 ∩ 𝑌) → (∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢 ∩ 𝑌)))
17	16	raleqbi1dv 3313	. . . . . . . 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 632	. . . . . . 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 247	. . . . . 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 3135	. . . . 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 24335	. . . . . . . . 9 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
22	21	ad2antrr 726	. . . . . . . 8 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → 𝐽 ∈ Top)
23		ssel2 3944	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ⊆ 𝑌 ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝑌)
24		ssel2 3944	. . . . . . . . . . . . . . . 16 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑋)
25		rpxr 12968	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑟 ∈ ℝ+ → 𝑟 ∈ ℝ*)
26	2	blopn 24395	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋 ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐶)𝑟) ∈ 𝐽)
27		eleq1a 2824	. . . . . . . . . . . . . . . . . . . 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 1118	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ*) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
30	25, 29	sylan2 593	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
31	30	rexlimdva 3135	. . . . . . . . . . . . . . . 16 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ 𝑋) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
32	24, 31	sylan2 593	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌)) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
33	32	anassrs 467	. . . . . . . . . . . . . 14 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑦 ∈ 𝑌) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
34	23, 33	sylan2 593	. . . . . . . . . . . . 13 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ 𝑦 ∈ 𝑥)) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
35	34	anassrs 467	. . . . . . . . . . . 12 ⊢ ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑌) ∧ 𝑦 ∈ 𝑥) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
36	35	rexlimdva 3135	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ 𝑥 ⊆ 𝑌) → (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧 ∈ 𝐽))
37	36	adantrd 491	. . . . . . . . . 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 717	. . . . . . . . 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 4034	. . . . . . . 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 22791	. . . . . . . 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 584	. . . . . . 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 7397	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑢 → (𝑦(ball‘𝐶)𝑟) = (𝑢(ball‘𝐶)𝑟))
43	42	ineq1d 4185	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑢 → ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) = ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌))
44	43	sseq1d 3981	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
45	44	rexbidv 3158	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑢 → (∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
46	45	rspccv 3588	. . . . . . . . . . . . . 14 ⊢ (∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → (𝑢 ∈ 𝑥 → ∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
47	46	ad2antll 729	. . . . . . . . . . . . 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 3943	. . . . . . . . . . . . . . 15 ⊢ (𝑥 ⊆ 𝑌 → (𝑢 ∈ 𝑥 → 𝑢 ∈ 𝑌))
49		ssel 3943	. . . . . . . . . . . . . . . 16 ⊢ (𝑌 ⊆ 𝑋 → (𝑢 ∈ 𝑌 → 𝑢 ∈ 𝑋))
50		blcntr 24308	. . . . . . . . . . . . . . . . . . . . 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 550	. . . . . . . . . . . . . . . . . . 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 1118	. . . . . . . . . . . . . . . . . 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 3147	. . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . 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 508	. . . . . . . . . . . . . . 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 508	. . . . . . . . . . . . . 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 717	. . . . . . . . . . . . 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 2815	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑢 → (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ↔ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))
61	44, 60	anbi12d 632	. . . . . . . . . . . . . . 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 3158	. . . . . . . . . . . . . 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 3591	. . . . . . . . . . . . 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 412	. . . . . . . . . . . 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 770	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) ∧ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → 𝑥 ⊆ 𝑌)
67	66	sseld 3948	. . . . . . . . . . 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 512	. . . . . . . . . 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 3933	. . . . . . . . . . . . . . 15 ⊢ (𝑢 ∈ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ↔ (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ∧ 𝑢 ∈ 𝑌))
70		ssel2 3944	. . . . . . . . . . . . . . 15 ⊢ ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌)) → 𝑢 ∈ 𝑥)
71	69, 70	sylan2br 595	. . . . . . . . . . . . . 14 ⊢ ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ∧ 𝑢 ∈ 𝑌)) → 𝑢 ∈ 𝑥)
72	71	expr 456	. . . . . . . . . . . . 13 ⊢ ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢 ∈ 𝑌 → 𝑢 ∈ 𝑥))
73	72	rexlimivw 3131	. . . . . . . . . . . 12 ⊢ (∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢 ∈ 𝑌 → 𝑢 ∈ 𝑥))
74	73	rexlimivw 3131	. . . . . . . . . . 11 ⊢ (∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢 ∈ 𝑌 → 𝑢 ∈ 𝑥))
75	74	imp 406	. . . . . . . . . 10 ⊢ ((∃𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ∧ 𝑢 ∈ 𝑌) → 𝑢 ∈ 𝑥)
76	68, 75	impbid1 225	. . . . . . . . 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 3933	. . . . . . . . . 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 4888	. . . . . . . . . . . 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 460	. . . . . . . . . . . . . 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 468	. . . . . . . . . . . . . 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 3168	. . . . . . . . . . . . . . . 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 3077	. . . . . . . . . . . . . . 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 3168	. . . . . . . . . . . . . . 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 276	. . . . . . . . . . . . . 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 297	. . . . . . . . . . . . 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 1848	. . . . . . . . . . . 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 7423	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦(ball‘𝐶)𝑟) ∈ V
88		ineq1 4179	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑧 = (𝑦(ball‘𝐶)𝑟) → (𝑧 ∩ 𝑌) = ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌))
89	88	sseq1d 3981	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑦(ball‘𝐶)𝑟) → ((𝑧 ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
90		eleq2 2818	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑧 = (𝑦(ball‘𝐶)𝑟) → (𝑢 ∈ 𝑧 ↔ 𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
91	89, 90	anbi12d 632	. . . . . . . . . . . . . . . . 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 3501	. . . . . . . . . . . . . . . 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 3077	. . . . . . . . . . . . . . 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 3265	. . . . . . . . . . . . . . 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 277	. . . . . . . . . . . . . 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 3077	. . . . . . . . . . . . 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 3265	. . . . . . . . . . . . 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 276	. . . . . . . . . . . 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 297	. . . . . . . . . . 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 624	. . . . . . . . . 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 276	. . . . . . . . 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 287	. . . . . . . 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 2728	. . . . . . 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 4179	. . . . . . . 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 3614	. . . . . . 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 584	. . . . . 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 412	. . . . 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 212	. . . 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 484	. . . . . . . . . . 11 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ 𝑌)
110	24, 109	elind 4166	. . . . . . . . . 10 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑦 ∈ 𝑌) → 𝑦 ∈ (𝑋 ∩ 𝑌))
111		metrest.1	. . . . . . . . . . . . . . 15 ⊢ 𝐷 = (𝐶 ↾ (𝑌 × 𝑌))
112	111	blres 24326	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) = ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌))
113	112	sseq1d 3981	. . . . . . . . . . . . 13 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌) ∧ 𝑟 ∈ ℝ*) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
114	113	3expa 1118	. . . . . . . . . . . 12 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌)) ∧ 𝑟 ∈ ℝ*) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
115	25, 114	sylan2 593	. . . . . . . . . . 11 ⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋 ∩ 𝑌)) ∧ 𝑟 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
116	115	rexbidva 3156	. . . . . . . . . 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 593	. . . . . . . . 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 467	. . . . . . . 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 593	. . . . . . 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 467	. . . . . 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 3155	. . . . 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 579	. . . 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 282	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌) ↔ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)))
124		id 22	. . . . 5 ⊢ (𝑌 ⊆ 𝑋 → 𝑌 ⊆ 𝑋)
125	2	mopnm 24339	. . . . 5 ⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝑋 ∈ 𝐽)
126		ssexg 5281	. . . . 5 ⊢ ((𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ 𝐽) → 𝑌 ∈ V)
127	124, 125, 126	syl2anr 597	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → 𝑌 ∈ V)
128		elrest 17397	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝑥 ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌)))
129	21, 127, 128	syl2an2r 685	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∈ (𝐽 ↾t 𝑌) ↔ ∃𝑢 ∈ 𝐽 𝑥 = (𝑢 ∩ 𝑌)))
130		xmetres2 24256	. . . . 5 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐶 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌))
131	111, 130	eqeltrid 2833	. . . 4 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → 𝐷 ∈ (∞Met‘𝑌))
132		metrest.4	. . . . 5 ⊢ 𝐾 = (MetOpen‘𝐷)
133	132	elmopn2 24340	. . . 4 ⊢ (𝐷 ∈ (∞Met‘𝑌) → (𝑥 ∈ 𝐾 ↔ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)))
134	131, 133	syl 17	. . 3 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∈ 𝐾 ↔ (𝑥 ⊆ 𝑌 ∧ ∀𝑦 ∈ 𝑥 ∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)))
135	123, 129, 134	3bitr4d 311	. 2 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∈ (𝐽 ↾t 𝑌) ↔ 𝑥 ∈ 𝐾))
136	135	eqrdv 2728	1 ⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) = 𝐾)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2708  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ∩ cin 3916   ⊆ wss 3917  ∪ cuni 4874   × cxp 5639   ↾ cres 5643  ‘cfv 6514  (class class class)co 7390  ℝ^*cxr 11214  ℝ⁺crp 12958   ↾_t crest 17390  ∞Metcxmet 21256  ballcbl 21258  MetOpencmopn 21261  Topctop 22787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152  ax-pre-sup 11153

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-sup 9400  df-inf 9401  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-n0 12450  df-z 12537  df-uz 12801  df-q 12915  df-rp 12959  df-xneg 13079  df-xadd 13080  df-xmul 13081  df-rest 17392  df-topgen 17413  df-psmet 21263  df-xmet 21264  df-bl 21266  df-mopn 21267  df-top 22788  df-topon 22805  df-bases 22840

This theorem is referenced by:  ressxms  24420  nrginvrcn  24587  resubmet  24697  tgioo2  24698  metdscn2  24753  divcnOLD  24764  divcn  24766  dfii3  24783  cncfcn  24810  metsscmetcld  25222  cmetss  25223  minveclem4a  25337  ftc1lem6  25955  ulmdvlem3  26318  abelth  26358  cxpcn3  26665  rlimcnp  26882  minvecolem4b  30814  minvecolem4  30816  hhsscms  31214  ftc1cnnc  37693