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Theorem metrest 22539
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 4029 . . . . . . . . . 10 (𝑢𝑌) ⊆ 𝑢
2 metrest.3 . . . . . . . . . . . . 13 𝐽 = (MetOpen‘𝐶)
32elmopn2 22460 . . . . . . . . . . . 12 (𝐶 ∈ (∞Met‘𝑋) → (𝑢𝐽 ↔ (𝑢𝑋 ∧ ∀𝑦𝑢𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)))
43simplbda 489 . . . . . . . . . . 11 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢𝐽) → ∀𝑦𝑢𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)
54adantlr 697 . . . . . . . . . 10 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑢𝐽) → ∀𝑦𝑢𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)
6 ssralv 3863 . . . . . . . . . 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 4034 . . . . . . . . . . 11 ((𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌))
98reximi 3198 . . . . . . . . . 10 (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌))
109ralimi 3140 . . . . . . . . 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 4030 . . . . . . . 8 (𝑢𝑌) ⊆ 𝑌
1311, 12jctil 511 . . . . . . 7 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑢𝐽) → ((𝑢𝑌) ⊆ 𝑌 ∧ ∀𝑦 ∈ (𝑢𝑌)∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌)))
14 sseq1 3823 . . . . . . . 8 (𝑥 = (𝑢𝑌) → (𝑥𝑌 ↔ (𝑢𝑌) ⊆ 𝑌))
15 sseq2 3824 . . . . . . . . . 10 (𝑥 = (𝑢𝑌) → (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌)))
1615rexbidv 3240 . . . . . . . . 9 (𝑥 = (𝑢𝑌) → (∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌)))
1716raleqbi1dv 3335 . . . . . . . 8 (𝑥 = (𝑢𝑌) → (∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∀𝑦 ∈ (𝑢𝑌)∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌)))
1814, 17anbi12d 618 . . . . . . 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 238 . . . . . 6 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑢𝐽) → (𝑥 = (𝑢𝑌) → (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
2019rexlimdva 3219 . . . . 5 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (∃𝑢𝐽 𝑥 = (𝑢𝑌) → (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
212mopntop 22455 . . . . . . . . 9 (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
2221ad2antrr 708 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → 𝐽 ∈ Top)
23 ssel2 3793 . . . . . . . . . . . . . 14 ((𝑥𝑌𝑦𝑥) → 𝑦𝑌)
24 ssel2 3793 . . . . . . . . . . . . . . . 16 ((𝑌𝑋𝑦𝑌) → 𝑦𝑋)
25 rpxr 12050 . . . . . . . . . . . . . . . . . 18 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
262blopn 22515 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) → (𝑦(ball‘𝐶)𝑟) ∈ 𝐽)
27 eleq1a 2880 . . . . . . . . . . . . . . . . . . . 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 1140 . . . . . . . . . . . . . . . . . 18 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋) ∧ 𝑟 ∈ ℝ*) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3025, 29sylan2 582 . . . . . . . . . . . . . . . . 17 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3130rexlimdva 3219 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3224, 31sylan2 582 . . . . . . . . . . . . . . 15 ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑌𝑋𝑦𝑌)) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3332anassrs 455 . . . . . . . . . . . . . 14 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑦𝑌) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3423, 33sylan2 582 . . . . . . . . . . . . 13 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌𝑦𝑥)) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3534anassrs 455 . . . . . . . . . . . 12 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) ∧ 𝑦𝑥) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3635rexlimdva 3219 . . . . . . . . . . 11 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) → (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3736adantrd 481 . . . . . . . . . 10 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) → ((∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥) → 𝑧𝐽))
3837adantrr 699 . . . . . . . . 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 3873 . . . . . . . 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 20912 . . . . . . . 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 575 . . . . . . 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 6877 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑢 → (𝑦(ball‘𝐶)𝑟) = (𝑢(ball‘𝐶)𝑟))
4342ineq1d 4012 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑢 → ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) = ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌))
4443sseq1d 3829 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑢 → (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
4544rexbidv 3240 . . . . . . . . . . . . . . 15 (𝑦 = 𝑢 → (∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
4645rspccv 3499 . . . . . . . . . . . . . 14 (∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → (𝑢𝑥 → ∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
4746ad2antll 711 . . . . . . . . . . . . 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 3792 . . . . . . . . . . . . . . 15 (𝑥𝑌 → (𝑢𝑥𝑢𝑌))
49 ssel 3792 . . . . . . . . . . . . . . . 16 (𝑌𝑋 → (𝑢𝑌𝑢𝑋))
50 blcntr 22428 . . . . . . . . . . . . . . . . . . . . 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 542 . . . . . . . . . . . . . . . . . . 19 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢𝑋𝑟 ∈ ℝ+) → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
53523expa 1140 . . . . . . . . . . . . . . . . . 18 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢𝑋) ∧ 𝑟 ∈ ℝ+) → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
5453reximdva 3204 . . . . . . . . . . . . . . . . 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 399 . . . . . . . . . . . . . . . 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 500 . . . . . . . . . . . . . . 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 500 . . . . . . . . . . . . . 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 699 . . . . . . . . . . . . 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 2871 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑢 → (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ↔ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))
6144, 60anbi12d 618 . . . . . . . . . . . . . . 15 (𝑦 = 𝑢 → ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ↔ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
6261rexbidv 3240 . . . . . . . . . . . . . 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 3502 . . . . . . . . . . . . 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 399 . . . . . . . . . . . 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 778 . . . . . . . . . . . 12 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → 𝑥𝑌)
6766sseld 3797 . . . . . . . . . . 11 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢𝑥𝑢𝑌))
6865, 67jcad 504 . . . . . . . . . 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 3995 . . . . . . . . . . . . . . 15 (𝑢 ∈ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ↔ (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ∧ 𝑢𝑌))
70 ssel2 3793 . . . . . . . . . . . . . . 15 ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌)) → 𝑢𝑥)
7169, 70sylan2br 584 . . . . . . . . . . . . . 14 ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ∧ 𝑢𝑌)) → 𝑢𝑥)
7271expr 446 . . . . . . . . . . . . 13 ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢𝑌𝑢𝑥))
7372rexlimivw 3217 . . . . . . . . . . . 12 (∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢𝑌𝑢𝑥))
7473rexlimivw 3217 . . . . . . . . . . 11 (∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢𝑌𝑢𝑥))
7574imp 395 . . . . . . . . . 10 ((∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ∧ 𝑢𝑌) → 𝑢𝑥)
7668, 75impbid1 216 . . . . . . . . 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 3995 . . . . . . . . . 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 4641 . . . . . . . . . . . 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 450 . . . . . . . . . . . . . 14 ((𝑢𝑧 ∧ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)) ↔ ((∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥) ∧ 𝑢𝑧))
80 anass 456 . . . . . . . . . . . . . 14 (((∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥) ∧ 𝑢𝑧) ↔ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
81 r19.41v 3277 . . . . . . . . . . . . . . . 16 (∃𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
8281rexbii 3229 . . . . . . . . . . . . . . 15 (∃𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ ∃𝑦𝑥 (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
83 r19.41v 3277 . . . . . . . . . . . . . . 15 (∃𝑦𝑥 (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
8482, 83bitr2i 267 . . . . . . . . . . . . . 14 ((∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ ∃𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
8579, 80, 843bitri 288 . . . . . . . . . . . . 13 ((𝑢𝑧 ∧ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)) ↔ ∃𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
8685exbii 1933 . . . . . . . . . . . 12 (∃𝑧(𝑢𝑧 ∧ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)) ↔ ∃𝑧𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
87 ovex 6902 . . . . . . . . . . . . . . . . 17 (𝑦(ball‘𝐶)𝑟) ∈ V
88 ineq1 4006 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑦(ball‘𝐶)𝑟) → (𝑧𝑌) = ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌))
8988sseq1d 3829 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑦(ball‘𝐶)𝑟) → ((𝑧𝑌) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
90 eleq2 2874 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑦(ball‘𝐶)𝑟) → (𝑢𝑧𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
9189, 90anbi12d 618 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑦(ball‘𝐶)𝑟) → (((𝑧𝑌) ⊆ 𝑥𝑢𝑧) ↔ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟))))
9287, 91ceqsexv 3436 . . . . . . . . . . . . . . . 16 (∃𝑧(𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
9392rexbii 3229 . . . . . . . . . . . . . . 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 3419 . . . . . . . . . . . . . . 15 (∃𝑟 ∈ ℝ+𝑧(𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ ∃𝑧𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
9593, 94bitr3i 268 . . . . . . . . . . . . . 14 (∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ↔ ∃𝑧𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
9695rexbii 3229 . . . . . . . . . . . . 13 (∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ↔ ∃𝑦𝑥𝑧𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
97 rexcom4 3419 . . . . . . . . . . . . 13 (∃𝑦𝑥𝑧𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ ∃𝑧𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
9896, 97bitr2i 267 . . . . . . . . . . . 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 288 . . . . . . . . . . 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 612 . . . . . . . . . 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 267 . . . . . . . . 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, 101syl6bb 278 . . . . . . . 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 2804 . . . . . . 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 4006 . . . . . . . 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 3520 . . . . . . 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 575 . . . . . 6 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → ∃𝑢𝐽 𝑥 = (𝑢𝑌))
107106ex 399 . . . . 5 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → ((𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥) → ∃𝑢𝐽 𝑥 = (𝑢𝑌)))
10820, 107impbid 203 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (∃𝑢𝐽 𝑥 = (𝑢𝑌) ↔ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
109 simpr 473 . . . . . . . . . . 11 ((𝑌𝑋𝑦𝑌) → 𝑦𝑌)
11024, 109elind 3997 . . . . . . . . . 10 ((𝑌𝑋𝑦𝑌) → 𝑦 ∈ (𝑋𝑌))
111 metrest.1 . . . . . . . . . . . . . . 15 𝐷 = (𝐶 ↾ (𝑌 × 𝑌))
112111blres 22446 . . . . . . . . . . . . . 14 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) = ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌))
113112sseq1d 3829 . . . . . . . . . . . . 13 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌) ∧ 𝑟 ∈ ℝ*) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
1141133expa 1140 . . . . . . . . . . . 12 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌)) ∧ 𝑟 ∈ ℝ*) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
11525, 114sylan2 582 . . . . . . . . . . 11 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌)) ∧ 𝑟 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
116115rexbidva 3237 . . . . . . . . . 10 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌)) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
117110, 116sylan2 582 . . . . . . . . 9 ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑌𝑋𝑦𝑌)) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
118117anassrs 455 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑦𝑌) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
11923, 118sylan2 582 . . . . . . 7 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌𝑦𝑥)) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
120119anassrs 455 . . . . . 6 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) ∧ 𝑦𝑥) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
121120ralbidva 3173 . . . . 5 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) → (∀𝑦𝑥𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
122121pm5.32da 570 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → ((𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥) ↔ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
123108, 122bitr4d 273 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (∃𝑢𝐽 𝑥 = (𝑢𝑌) ↔ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)))
12421adantr 468 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → 𝐽 ∈ Top)
125 id 22 . . . . 5 (𝑌𝑋𝑌𝑋)
1262mopnm 22459 . . . . 5 (𝐶 ∈ (∞Met‘𝑋) → 𝑋𝐽)
127 ssexg 4999 . . . . 5 ((𝑌𝑋𝑋𝐽) → 𝑌 ∈ V)
128125, 126, 127syl2anr 586 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → 𝑌 ∈ V)
129 elrest 16289 . . . 4 ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝑥 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑥 = (𝑢𝑌)))
130124, 128, 129syl2anc 575 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝑥 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑥 = (𝑢𝑌)))
131 xmetres2 22376 . . . . 5 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝐶 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌))
132111, 131syl5eqel 2889 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → 𝐷 ∈ (∞Met‘𝑌))
133 metrest.4 . . . . 5 𝐾 = (MetOpen‘𝐷)
134133elmopn2 22460 . . . 4 (𝐷 ∈ (∞Met‘𝑌) → (𝑥𝐾 ↔ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)))
135132, 134syl 17 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝑥𝐾 ↔ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)))
136123, 130, 1353bitr4d 302 . 2 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝑥 ∈ (𝐽t 𝑌) ↔ 𝑥𝐾))
137136eqrdv 2804 1 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2156  {cab 2792  wral 3096  wrex 3097  Vcvv 3391  cin 3768  wss 3769   cuni 4630   × cxp 5309  cres 5313  cfv 6097  (class class class)co 6870  *cxr 10354  +crp 12042  t crest 16282  ∞Metcxmt 19935  ballcbl 19937  MetOpencmopn 19940  Topctop 20908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175  ax-cnex 10273  ax-resscn 10274  ax-1cn 10275  ax-icn 10276  ax-addcl 10277  ax-addrcl 10278  ax-mulcl 10279  ax-mulrcl 10280  ax-mulcom 10281  ax-addass 10282  ax-mulass 10283  ax-distr 10284  ax-i2m1 10285  ax-1ne0 10286  ax-1rid 10287  ax-rnegex 10288  ax-rrecex 10289  ax-cnre 10290  ax-pre-lttri 10291  ax-pre-lttrn 10292  ax-pre-ltadd 10293  ax-pre-mulgt0 10294  ax-pre-sup 10295
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-nel 3082  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-riota 6831  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-om 7292  df-1st 7394  df-2nd 7395  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-er 7975  df-map 8090  df-en 8189  df-dom 8190  df-sdom 8191  df-sup 8583  df-inf 8584  df-pnf 10357  df-mnf 10358  df-xr 10359  df-ltxr 10360  df-le 10361  df-sub 10549  df-neg 10550  df-div 10966  df-nn 11302  df-2 11360  df-n0 11556  df-z 11640  df-uz 11901  df-q 12004  df-rp 12043  df-xneg 12158  df-xadd 12159  df-xmul 12160  df-rest 16284  df-topgen 16305  df-psmet 19942  df-xmet 19943  df-bl 19945  df-mopn 19946  df-top 20909  df-topon 20926  df-bases 20961
This theorem is referenced by:  ressxms  22540  nrginvrcn  22706  resubmet  22815  tgioo2  22816  metdscn2  22870  divcn  22881  dfii3  22896  cncfcn  22922  cmetss  23323  minveclem4a  23412  ftc1lem6  24017  ulmdvlem3  24369  abelth  24408  cxpcn3  24702  rlimcnp  24905  minvecolem4b  28061  minvecolem4  28063  hhsscms  28463  ftc1cnnc  33794
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