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Theorem metrest 24538
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) ∧ 𝑌𝑋) → (𝐽t 𝑌) = 𝐾)

Proof of Theorem metrest
Dummy variables 𝑢 𝑟 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 4236 . . . . . . . . . 10 (𝑢𝑌) ⊆ 𝑢
2 metrest.3 . . . . . . . . . . . . 13 𝐽 = (MetOpen‘𝐶)
32elmopn2 24456 . . . . . . . . . . . 12 (𝐶 ∈ (∞Met‘𝑋) → (𝑢𝐽 ↔ (𝑢𝑋 ∧ ∀𝑦𝑢𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)))
43simplbda 499 . . . . . . . . . . 11 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢𝐽) → ∀𝑦𝑢𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)
54adantlr 715 . . . . . . . . . 10 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑢𝐽) → ∀𝑦𝑢𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)
6 ssralv 4051 . . . . . . . . . 10 ((𝑢𝑌) ⊆ 𝑢 → (∀𝑦𝑢𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ∀𝑦 ∈ (𝑢𝑌)∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢))
71, 5, 6mpsyl 68 . . . . . . . . 9 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑢𝐽) → ∀𝑦 ∈ (𝑢𝑌)∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)
8 ssrin 4241 . . . . . . . . . . 11 ((𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌))
98reximi 3083 . . . . . . . . . 10 (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌))
109ralimi 3082 . . . . . . . . 9 (∀𝑦 ∈ (𝑢𝑌)∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ∀𝑦 ∈ (𝑢𝑌)∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌))
117, 10syl 17 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑢𝐽) → ∀𝑦 ∈ (𝑢𝑌)∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌))
12 inss2 4237 . . . . . . . 8 (𝑢𝑌) ⊆ 𝑌
1311, 12jctil 519 . . . . . . 7 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑢𝐽) → ((𝑢𝑌) ⊆ 𝑌 ∧ ∀𝑦 ∈ (𝑢𝑌)∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌)))
14 sseq1 4008 . . . . . . . 8 (𝑥 = (𝑢𝑌) → (𝑥𝑌 ↔ (𝑢𝑌) ⊆ 𝑌))
15 sseq2 4009 . . . . . . . . . 10 (𝑥 = (𝑢𝑌) → (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌)))
1615rexbidv 3178 . . . . . . . . 9 (𝑥 = (𝑢𝑌) → (∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌)))
1716raleqbi1dv 3337 . . . . . . . 8 (𝑥 = (𝑢𝑌) → (∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∀𝑦 ∈ (𝑢𝑌)∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌)))
1814, 17anbi12d 632 . . . . . . 7 (𝑥 = (𝑢𝑌) → ((𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥) ↔ ((𝑢𝑌) ⊆ 𝑌 ∧ ∀𝑦 ∈ (𝑢𝑌)∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌))))
1913, 18syl5ibrcom 247 . . . . . 6 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑢𝐽) → (𝑥 = (𝑢𝑌) → (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
2019rexlimdva 3154 . . . . 5 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (∃𝑢𝐽 𝑥 = (𝑢𝑌) → (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
212mopntop 24451 . . . . . . . . 9 (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
2221ad2antrr 726 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → 𝐽 ∈ Top)
23 ssel2 3977 . . . . . . . . . . . . . 14 ((𝑥𝑌𝑦𝑥) → 𝑦𝑌)
24 ssel2 3977 . . . . . . . . . . . . . . . 16 ((𝑌𝑋𝑦𝑌) → 𝑦𝑋)
25 rpxr 13045 . . . . . . . . . . . . . . . . . 18 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
262blopn 24514 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) → (𝑦(ball‘𝐶)𝑟) ∈ 𝐽)
27 eleq1a 2835 . . . . . . . . . . . . . . . . . . . 20 ((𝑦(ball‘𝐶)𝑟) ∈ 𝐽 → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
2826, 27syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
29283expa 1118 . . . . . . . . . . . . . . . . . 18 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋) ∧ 𝑟 ∈ ℝ*) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3025, 29sylan2 593 . . . . . . . . . . . . . . . . 17 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3130rexlimdva 3154 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3224, 31sylan2 593 . . . . . . . . . . . . . . 15 ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑌𝑋𝑦𝑌)) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3332anassrs 467 . . . . . . . . . . . . . 14 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑦𝑌) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3423, 33sylan2 593 . . . . . . . . . . . . 13 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌𝑦𝑥)) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3534anassrs 467 . . . . . . . . . . . 12 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) ∧ 𝑦𝑥) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3635rexlimdva 3154 . . . . . . . . . . 11 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) → (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3736adantrd 491 . . . . . . . . . 10 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) → ((∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥) → 𝑧𝐽))
3837adantrr 717 . . . . . . . . 9 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → ((∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥) → 𝑧𝐽))
3938abssdv 4067 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → {𝑧 ∣ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)} ⊆ 𝐽)
40 uniopn 22904 . . . . . . . 8 ((𝐽 ∈ Top ∧ {𝑧 ∣ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)} ⊆ 𝐽) → {𝑧 ∣ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)} ∈ 𝐽)
4122, 39, 40syl2anc 584 . . . . . . 7 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → {𝑧 ∣ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)} ∈ 𝐽)
42 oveq1 7439 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑢 → (𝑦(ball‘𝐶)𝑟) = (𝑢(ball‘𝐶)𝑟))
4342ineq1d 4218 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑢 → ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) = ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌))
4443sseq1d 4014 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑢 → (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
4544rexbidv 3178 . . . . . . . . . . . . . . 15 (𝑦 = 𝑢 → (∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
4645rspccv 3618 . . . . . . . . . . . . . 14 (∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → (𝑢𝑥 → ∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
4746ad2antll 729 . . . . . . . . . . . . 13 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢𝑥 → ∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
48 ssel 3976 . . . . . . . . . . . . . . 15 (𝑥𝑌 → (𝑢𝑥𝑢𝑌))
49 ssel 3976 . . . . . . . . . . . . . . . 16 (𝑌𝑋 → (𝑢𝑌𝑢𝑋))
50 blcntr 24424 . . . . . . . . . . . . . . . . . . . . 21 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢𝑋𝑟 ∈ ℝ+) → 𝑢 ∈ (𝑢(ball‘𝐶)𝑟))
5150a1d 25 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢𝑋𝑟 ∈ ℝ+) → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))
5251ancld 550 . . . . . . . . . . . . . . . . . . 19 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢𝑋𝑟 ∈ ℝ+) → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
53523expa 1118 . . . . . . . . . . . . . . . . . 18 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢𝑋) ∧ 𝑟 ∈ ℝ+) → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
5453reximdva 3167 . . . . . . . . . . . . . . . . 17 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢𝑋) → (∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → ∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
5554ex 412 . . . . . . . . . . . . . . . 16 (𝐶 ∈ (∞Met‘𝑋) → (𝑢𝑋 → (∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → ∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))))
5649, 55sylan9r 508 . . . . . . . . . . . . . . 15 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝑢𝑌 → (∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → ∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))))
5748, 56sylan9r 508 . . . . . . . . . . . . . 14 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) → (𝑢𝑥 → (∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → ∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))))
5857adantrr 717 . . . . . . . . . . . . 13 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢𝑥 → (∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → ∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))))
5947, 58mpdd 43 . . . . . . . . . . . 12 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢𝑥 → ∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
6042eleq2d 2826 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑢 → (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ↔ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))
6144, 60anbi12d 632 . . . . . . . . . . . . . . 15 (𝑦 = 𝑢 → ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ↔ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
6261rexbidv 3178 . . . . . . . . . . . . . 14 (𝑦 = 𝑢 → (∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ↔ ∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
6362rspcev 3621 . . . . . . . . . . . . 13 ((𝑢𝑥 ∧ ∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟))) → ∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
6463ex 412 . . . . . . . . . . . 12 (𝑢𝑥 → (∃𝑟 ∈ ℝ+ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟)) → ∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟))))
6559, 64sylcom 30 . . . . . . . . . . 11 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢𝑥 → ∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟))))
66 simprl 770 . . . . . . . . . . . 12 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → 𝑥𝑌)
6766sseld 3981 . . . . . . . . . . 11 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢𝑥𝑢𝑌))
6865, 67jcad 512 . . . . . . . . . 10 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢𝑥 → (∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ∧ 𝑢𝑌)))
69 elin 3966 . . . . . . . . . . . . . . 15 (𝑢 ∈ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ↔ (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ∧ 𝑢𝑌))
70 ssel2 3977 . . . . . . . . . . . . . . 15 ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌)) → 𝑢𝑥)
7169, 70sylan2br 595 . . . . . . . . . . . . . 14 ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ∧ 𝑢𝑌)) → 𝑢𝑥)
7271expr 456 . . . . . . . . . . . . 13 ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢𝑌𝑢𝑥))
7372rexlimivw 3150 . . . . . . . . . . . 12 (∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢𝑌𝑢𝑥))
7473rexlimivw 3150 . . . . . . . . . . 11 (∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢𝑌𝑢𝑥))
7574imp 406 . . . . . . . . . 10 ((∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ∧ 𝑢𝑌) → 𝑢𝑥)
7668, 75impbid1 225 . . . . . . . . 9 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢𝑥 ↔ (∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ∧ 𝑢𝑌)))
77 elin 3966 . . . . . . . . . 10 (𝑢 ∈ ( {𝑧 ∣ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)} ∩ 𝑌) ↔ (𝑢 {𝑧 ∣ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)} ∧ 𝑢𝑌))
78 eluniab 4920 . . . . . . . . . . . 12 (𝑢 {𝑧 ∣ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)} ↔ ∃𝑧(𝑢𝑧 ∧ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)))
79 ancom 460 . . . . . . . . . . . . . 14 ((𝑢𝑧 ∧ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)) ↔ ((∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥) ∧ 𝑢𝑧))
80 anass 468 . . . . . . . . . . . . . 14 (((∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥) ∧ 𝑢𝑧) ↔ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
81 r19.41v 3188 . . . . . . . . . . . . . . . 16 (∃𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
8281rexbii 3093 . . . . . . . . . . . . . . 15 (∃𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ ∃𝑦𝑥 (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
83 r19.41v 3188 . . . . . . . . . . . . . . 15 (∃𝑦𝑥 (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
8482, 83bitr2i 276 . . . . . . . . . . . . . 14 ((∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ ∃𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
8579, 80, 843bitri 297 . . . . . . . . . . . . 13 ((𝑢𝑧 ∧ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)) ↔ ∃𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
8685exbii 1847 . . . . . . . . . . . 12 (∃𝑧(𝑢𝑧 ∧ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)) ↔ ∃𝑧𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
87 ovex 7465 . . . . . . . . . . . . . . . . 17 (𝑦(ball‘𝐶)𝑟) ∈ V
88 ineq1 4212 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑦(ball‘𝐶)𝑟) → (𝑧𝑌) = ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌))
8988sseq1d 4014 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑦(ball‘𝐶)𝑟) → ((𝑧𝑌) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
90 eleq2 2829 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑦(ball‘𝐶)𝑟) → (𝑢𝑧𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
9189, 90anbi12d 632 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑦(ball‘𝐶)𝑟) → (((𝑧𝑌) ⊆ 𝑥𝑢𝑧) ↔ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟))))
9287, 91ceqsexv 3531 . . . . . . . . . . . . . . . 16 (∃𝑧(𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
9392rexbii 3093 . . . . . . . . . . . . . . 15 (∃𝑟 ∈ ℝ+𝑧(𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ ∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
94 rexcom4 3287 . . . . . . . . . . . . . . 15 (∃𝑟 ∈ ℝ+𝑧(𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ ∃𝑧𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
9593, 94bitr3i 277 . . . . . . . . . . . . . 14 (∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ↔ ∃𝑧𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
9695rexbii 3093 . . . . . . . . . . . . 13 (∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ↔ ∃𝑦𝑥𝑧𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
97 rexcom4 3287 . . . . . . . . . . . . 13 (∃𝑦𝑥𝑧𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ ∃𝑧𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
9896, 97bitr2i 276 . . . . . . . . . . . 12 (∃𝑧𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ ∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
9978, 86, 983bitri 297 . . . . . . . . . . 11 (𝑢 {𝑧 ∣ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)} ↔ ∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
10099anbi1i 624 . . . . . . . . . 10 ((𝑢 {𝑧 ∣ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)} ∧ 𝑢𝑌) ↔ (∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ∧ 𝑢𝑌))
10177, 100bitr2i 276 . . . . . . . . 9 ((∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ∧ 𝑢𝑌) ↔ 𝑢 ∈ ( {𝑧 ∣ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)} ∩ 𝑌))
10276, 101bitrdi 287 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢𝑥𝑢 ∈ ( {𝑧 ∣ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)} ∩ 𝑌)))
103102eqrdv 2734 . . . . . . 7 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → 𝑥 = ( {𝑧 ∣ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)} ∩ 𝑌))
104 ineq1 4212 . . . . . . . 8 (𝑢 = {𝑧 ∣ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)} → (𝑢𝑌) = ( {𝑧 ∣ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)} ∩ 𝑌))
105104rspceeqv 3644 . . . . . . 7 (( {𝑧 ∣ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)} ∈ 𝐽𝑥 = ( {𝑧 ∣ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)} ∩ 𝑌)) → ∃𝑢𝐽 𝑥 = (𝑢𝑌))
10641, 103, 105syl2anc 584 . . . . . 6 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → ∃𝑢𝐽 𝑥 = (𝑢𝑌))
107106ex 412 . . . . 5 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → ((𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥) → ∃𝑢𝐽 𝑥 = (𝑢𝑌)))
10820, 107impbid 212 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (∃𝑢𝐽 𝑥 = (𝑢𝑌) ↔ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
109 simpr 484 . . . . . . . . . . 11 ((𝑌𝑋𝑦𝑌) → 𝑦𝑌)
11024, 109elind 4199 . . . . . . . . . 10 ((𝑌𝑋𝑦𝑌) → 𝑦 ∈ (𝑋𝑌))
111 metrest.1 . . . . . . . . . . . . . . 15 𝐷 = (𝐶 ↾ (𝑌 × 𝑌))
112111blres 24442 . . . . . . . . . . . . . 14 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) = ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌))
113112sseq1d 4014 . . . . . . . . . . . . 13 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌) ∧ 𝑟 ∈ ℝ*) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
1141133expa 1118 . . . . . . . . . . . 12 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌)) ∧ 𝑟 ∈ ℝ*) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
11525, 114sylan2 593 . . . . . . . . . . 11 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌)) ∧ 𝑟 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
116115rexbidva 3176 . . . . . . . . . 10 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌)) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
117110, 116sylan2 593 . . . . . . . . 9 ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑌𝑋𝑦𝑌)) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
118117anassrs 467 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑦𝑌) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
11923, 118sylan2 593 . . . . . . 7 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌𝑦𝑥)) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
120119anassrs 467 . . . . . 6 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) ∧ 𝑦𝑥) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
121120ralbidva 3175 . . . . 5 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) → (∀𝑦𝑥𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
122121pm5.32da 579 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → ((𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥) ↔ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
123108, 122bitr4d 282 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (∃𝑢𝐽 𝑥 = (𝑢𝑌) ↔ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)))
124 id 22 . . . . 5 (𝑌𝑋𝑌𝑋)
1252mopnm 24455 . . . . 5 (𝐶 ∈ (∞Met‘𝑋) → 𝑋𝐽)
126 ssexg 5322 . . . . 5 ((𝑌𝑋𝑋𝐽) → 𝑌 ∈ V)
127124, 125, 126syl2anr 597 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → 𝑌 ∈ V)
128 elrest 17473 . . . 4 ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝑥 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑥 = (𝑢𝑌)))
12921, 127, 128syl2an2r 685 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝑥 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑥 = (𝑢𝑌)))
130 xmetres2 24372 . . . . 5 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝐶 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌))
131111, 130eqeltrid 2844 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → 𝐷 ∈ (∞Met‘𝑌))
132 metrest.4 . . . . 5 𝐾 = (MetOpen‘𝐷)
133132elmopn2 24456 . . . 4 (𝐷 ∈ (∞Met‘𝑌) → (𝑥𝐾 ↔ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)))
134131, 133syl 17 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝑥𝐾 ↔ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)))
135123, 129, 1343bitr4d 311 . 2 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝑥 ∈ (𝐽t 𝑌) ↔ 𝑥𝐾))
136135eqrdv 2734 1 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wex 1778  wcel 2107  {cab 2713  wral 3060  wrex 3069  Vcvv 3479  cin 3949  wss 3950   cuni 4906   × cxp 5682  cres 5686  cfv 6560  (class class class)co 7432  *cxr 11295  +crp 13035  t crest 17466  ∞Metcxmet 21350  ballcbl 21352  MetOpencmopn 21355  Topctop 22900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233  ax-pre-sup 11234
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-sup 9483  df-inf 9484  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-n0 12529  df-z 12616  df-uz 12880  df-q 12992  df-rp 13036  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-rest 17468  df-topgen 17489  df-psmet 21357  df-xmet 21358  df-bl 21360  df-mopn 21361  df-top 22901  df-topon 22918  df-bases 22954
This theorem is referenced by:  ressxms  24539  nrginvrcn  24714  resubmet  24824  tgioo2  24825  metdscn2  24880  divcnOLD  24891  divcn  24893  dfii3  24910  cncfcn  24937  metsscmetcld  25350  cmetss  25351  minveclem4a  25465  ftc1lem6  26083  ulmdvlem3  26446  abelth  26486  cxpcn3  26792  rlimcnp  27009  minvecolem4b  30898  minvecolem4  30900  hhsscms  31298  ftc1cnnc  37700
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