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Theorem metrest 24477
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 4227 . . . . . . . . . 10 (𝑢𝑌) ⊆ 𝑢
2 metrest.3 . . . . . . . . . . . . 13 𝐽 = (MetOpen‘𝐶)
32elmopn2 24395 . . . . . . . . . . . 12 (𝐶 ∈ (∞Met‘𝑋) → (𝑢𝐽 ↔ (𝑢𝑋 ∧ ∀𝑦𝑢𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)))
43simplbda 498 . . . . . . . . . . 11 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢𝐽) → ∀𝑦𝑢𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)
54adantlr 713 . . . . . . . . . 10 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑢𝐽) → ∀𝑦𝑢𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)
6 ssralv 4045 . . . . . . . . . 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 4232 . . . . . . . . . . 11 ((𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌))
98reximi 3073 . . . . . . . . . 10 (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌))
109ralimi 3072 . . . . . . . . 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 4228 . . . . . . . 8 (𝑢𝑌) ⊆ 𝑌
1311, 12jctil 518 . . . . . . 7 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑢𝐽) → ((𝑢𝑌) ⊆ 𝑌 ∧ ∀𝑦 ∈ (𝑢𝑌)∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌)))
14 sseq1 4002 . . . . . . . 8 (𝑥 = (𝑢𝑌) → (𝑥𝑌 ↔ (𝑢𝑌) ⊆ 𝑌))
15 sseq2 4003 . . . . . . . . . 10 (𝑥 = (𝑢𝑌) → (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌)))
1615rexbidv 3168 . . . . . . . . 9 (𝑥 = (𝑢𝑌) → (∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌)))
1716raleqbi1dv 3322 . . . . . . . 8 (𝑥 = (𝑢𝑌) → (∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∀𝑦 ∈ (𝑢𝑌)∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌)))
1814, 17anbi12d 630 . . . . . . 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 246 . . . . . 6 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑢𝐽) → (𝑥 = (𝑢𝑌) → (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
2019rexlimdva 3144 . . . . 5 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (∃𝑢𝐽 𝑥 = (𝑢𝑌) → (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
212mopntop 24390 . . . . . . . . 9 (𝐶 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top)
2221ad2antrr 724 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → 𝐽 ∈ Top)
23 ssel2 3971 . . . . . . . . . . . . . 14 ((𝑥𝑌𝑦𝑥) → 𝑦𝑌)
24 ssel2 3971 . . . . . . . . . . . . . . . 16 ((𝑌𝑋𝑦𝑌) → 𝑦𝑋)
25 rpxr 13018 . . . . . . . . . . . . . . . . . 18 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
262blopn 24453 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) → (𝑦(ball‘𝐶)𝑟) ∈ 𝐽)
27 eleq1a 2820 . . . . . . . . . . . . . . . . . . . 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 1115 . . . . . . . . . . . . . . . . . 18 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋) ∧ 𝑟 ∈ ℝ*) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3025, 29sylan2 591 . . . . . . . . . . . . . . . . 17 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3130rexlimdva 3144 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3224, 31sylan2 591 . . . . . . . . . . . . . . 15 ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑌𝑋𝑦𝑌)) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3332anassrs 466 . . . . . . . . . . . . . 14 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑦𝑌) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3423, 33sylan2 591 . . . . . . . . . . . . 13 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌𝑦𝑥)) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3534anassrs 466 . . . . . . . . . . . 12 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) ∧ 𝑦𝑥) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3635rexlimdva 3144 . . . . . . . . . . 11 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) → (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3736adantrd 490 . . . . . . . . . 10 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) → ((∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥) → 𝑧𝐽))
3837adantrr 715 . . . . . . . . 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 4061 . . . . . . . 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 22843 . . . . . . . 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 582 . . . . . . 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 7426 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑢 → (𝑦(ball‘𝐶)𝑟) = (𝑢(ball‘𝐶)𝑟))
4342ineq1d 4209 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑢 → ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) = ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌))
4443sseq1d 4008 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑢 → (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
4544rexbidv 3168 . . . . . . . . . . . . . . 15 (𝑦 = 𝑢 → (∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
4645rspccv 3603 . . . . . . . . . . . . . 14 (∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → (𝑢𝑥 → ∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
4746ad2antll 727 . . . . . . . . . . . . 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 3970 . . . . . . . . . . . . . . 15 (𝑥𝑌 → (𝑢𝑥𝑢𝑌))
49 ssel 3970 . . . . . . . . . . . . . . . 16 (𝑌𝑋 → (𝑢𝑌𝑢𝑋))
50 blcntr 24363 . . . . . . . . . . . . . . . . . . . . 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 549 . . . . . . . . . . . . . . . . . . 19 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢𝑋𝑟 ∈ ℝ+) → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
53523expa 1115 . . . . . . . . . . . . . . . . . 18 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑢𝑋) ∧ 𝑟 ∈ ℝ+) → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 → (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
5453reximdva 3157 . . . . . . . . . . . . . . . . 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 411 . . . . . . . . . . . . . . . 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 507 . . . . . . . . . . . . . . 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 507 . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . 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 2811 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑢 → (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ↔ 𝑢 ∈ (𝑢(ball‘𝐶)𝑟)))
6144, 60anbi12d 630 . . . . . . . . . . . . . . 15 (𝑦 = 𝑢 → ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ↔ (((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑢(ball‘𝐶)𝑟))))
6261rexbidv 3168 . . . . . . . . . . . . . 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 3606 . . . . . . . . . . . . 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 411 . . . . . . . . . . . 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 769 . . . . . . . . . . . 12 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → 𝑥𝑌)
6766sseld 3975 . . . . . . . . . . 11 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢𝑥𝑢𝑌))
6865, 67jcad 511 . . . . . . . . . 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 3960 . . . . . . . . . . . . . . 15 (𝑢 ∈ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ↔ (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ∧ 𝑢𝑌))
70 ssel2 3971 . . . . . . . . . . . . . . 15 ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌)) → 𝑢𝑥)
7169, 70sylan2br 593 . . . . . . . . . . . . . 14 ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ∧ 𝑢𝑌)) → 𝑢𝑥)
7271expr 455 . . . . . . . . . . . . 13 ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢𝑌𝑢𝑥))
7372rexlimivw 3140 . . . . . . . . . . . 12 (∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢𝑌𝑢𝑥))
7473rexlimivw 3140 . . . . . . . . . . 11 (∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢𝑌𝑢𝑥))
7574imp 405 . . . . . . . . . 10 ((∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ∧ 𝑢𝑌) → 𝑢𝑥)
7668, 75impbid1 224 . . . . . . . . 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 3960 . . . . . . . . . 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 4923 . . . . . . . . . . . 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 459 . . . . . . . . . . . . . 14 ((𝑢𝑧 ∧ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)) ↔ ((∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥) ∧ 𝑢𝑧))
80 anass 467 . . . . . . . . . . . . . 14 (((∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥) ∧ 𝑢𝑧) ↔ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
81 r19.41v 3178 . . . . . . . . . . . . . . . 16 (∃𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
8281rexbii 3083 . . . . . . . . . . . . . . 15 (∃𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ ∃𝑦𝑥 (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
83 r19.41v 3178 . . . . . . . . . . . . . . 15 (∃𝑦𝑥 (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
8482, 83bitr2i 275 . . . . . . . . . . . . . 14 ((∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ ∃𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
8579, 80, 843bitri 296 . . . . . . . . . . . . 13 ((𝑢𝑧 ∧ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)) ↔ ∃𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
8685exbii 1842 . . . . . . . . . . . 12 (∃𝑧(𝑢𝑧 ∧ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)) ↔ ∃𝑧𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
87 ovex 7452 . . . . . . . . . . . . . . . . 17 (𝑦(ball‘𝐶)𝑟) ∈ V
88 ineq1 4203 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑦(ball‘𝐶)𝑟) → (𝑧𝑌) = ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌))
8988sseq1d 4008 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑦(ball‘𝐶)𝑟) → ((𝑧𝑌) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
90 eleq2 2814 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑦(ball‘𝐶)𝑟) → (𝑢𝑧𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
9189, 90anbi12d 630 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑦(ball‘𝐶)𝑟) → (((𝑧𝑌) ⊆ 𝑥𝑢𝑧) ↔ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟))))
9287, 91ceqsexv 3514 . . . . . . . . . . . . . . . 16 (∃𝑧(𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
9392rexbii 3083 . . . . . . . . . . . . . . 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 3275 . . . . . . . . . . . . . . 15 (∃𝑟 ∈ ℝ+𝑧(𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ ∃𝑧𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
9593, 94bitr3i 276 . . . . . . . . . . . . . 14 (∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ↔ ∃𝑧𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
9695rexbii 3083 . . . . . . . . . . . . 13 (∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ↔ ∃𝑦𝑥𝑧𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
97 rexcom4 3275 . . . . . . . . . . . . 13 (∃𝑦𝑥𝑧𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ ∃𝑧𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
9896, 97bitr2i 275 . . . . . . . . . . . 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 296 . . . . . . . . . . 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 622 . . . . . . . . . 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 275 . . . . . . . . 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 286 . . . . . . . 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 2723 . . . . . . 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 4203 . . . . . . . 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 3628 . . . . . . 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 582 . . . . . 6 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → ∃𝑢𝐽 𝑥 = (𝑢𝑌))
107106ex 411 . . . . 5 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → ((𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥) → ∃𝑢𝐽 𝑥 = (𝑢𝑌)))
10820, 107impbid 211 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (∃𝑢𝐽 𝑥 = (𝑢𝑌) ↔ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
109 simpr 483 . . . . . . . . . . 11 ((𝑌𝑋𝑦𝑌) → 𝑦𝑌)
11024, 109elind 4192 . . . . . . . . . 10 ((𝑌𝑋𝑦𝑌) → 𝑦 ∈ (𝑋𝑌))
111 metrest.1 . . . . . . . . . . . . . . 15 𝐷 = (𝐶 ↾ (𝑌 × 𝑌))
112111blres 24381 . . . . . . . . . . . . . 14 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) = ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌))
113112sseq1d 4008 . . . . . . . . . . . . 13 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌) ∧ 𝑟 ∈ ℝ*) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
1141133expa 1115 . . . . . . . . . . . 12 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌)) ∧ 𝑟 ∈ ℝ*) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
11525, 114sylan2 591 . . . . . . . . . . 11 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌)) ∧ 𝑟 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
116115rexbidva 3166 . . . . . . . . . 10 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌)) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
117110, 116sylan2 591 . . . . . . . . 9 ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑌𝑋𝑦𝑌)) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
118117anassrs 466 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑦𝑌) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
11923, 118sylan2 591 . . . . . . 7 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌𝑦𝑥)) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
120119anassrs 466 . . . . . 6 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) ∧ 𝑦𝑥) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
121120ralbidva 3165 . . . . 5 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) → (∀𝑦𝑥𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
122121pm5.32da 577 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → ((𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥) ↔ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
123108, 122bitr4d 281 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (∃𝑢𝐽 𝑥 = (𝑢𝑌) ↔ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥)))
124 id 22 . . . . 5 (𝑌𝑋𝑌𝑋)
1252mopnm 24394 . . . . 5 (𝐶 ∈ (∞Met‘𝑋) → 𝑋𝐽)
126 ssexg 5324 . . . . 5 ((𝑌𝑋𝑋𝐽) → 𝑌 ∈ V)
127124, 125, 126syl2anr 595 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → 𝑌 ∈ V)
128 elrest 17412 . . . 4 ((𝐽 ∈ Top ∧ 𝑌 ∈ V) → (𝑥 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑥 = (𝑢𝑌)))
12921, 127, 128syl2an2r 683 . . 3 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝑥 ∈ (𝐽t 𝑌) ↔ ∃𝑢𝐽 𝑥 = (𝑢𝑌)))
130 xmetres2 24311 . . . . 5 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝐶 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌))
131111, 130eqeltrid 2829 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → 𝐷 ∈ (∞Met‘𝑌))
132 metrest.4 . . . . 5 𝐾 = (MetOpen‘𝐷)
133132elmopn2 24395 . . . 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 310 . 2 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝑥 ∈ (𝐽t 𝑌) ↔ 𝑥𝐾))
136135eqrdv 2723 1 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wex 1773  wcel 2098  {cab 2702  wral 3050  wrex 3059  Vcvv 3461  cin 3943  wss 3944   cuni 4909   × cxp 5676  cres 5680  cfv 6549  (class class class)co 7419  *cxr 11279  +crp 13009  t crest 17405  ∞Metcxmet 21281  ballcbl 21283  MetOpencmopn 21286  Topctop 22839
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5365  ax-pr 5429  ax-un 7741  ax-cnex 11196  ax-resscn 11197  ax-1cn 11198  ax-icn 11199  ax-addcl 11200  ax-addrcl 11201  ax-mulcl 11202  ax-mulrcl 11203  ax-mulcom 11204  ax-addass 11205  ax-mulass 11206  ax-distr 11207  ax-i2m1 11208  ax-1ne0 11209  ax-1rid 11210  ax-rnegex 11211  ax-rrecex 11212  ax-cnre 11213  ax-pre-lttri 11214  ax-pre-lttrn 11215  ax-pre-ltadd 11216  ax-pre-mulgt0 11217  ax-pre-sup 11218
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-nel 3036  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-iun 4999  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6307  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-f1 6554  df-fo 6555  df-f1o 6556  df-fv 6557  df-riota 7375  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7872  df-1st 7994  df-2nd 7995  df-frecs 8287  df-wrecs 8318  df-recs 8392  df-rdg 8431  df-er 8725  df-map 8847  df-en 8965  df-dom 8966  df-sdom 8967  df-sup 9467  df-inf 9468  df-pnf 11282  df-mnf 11283  df-xr 11284  df-ltxr 11285  df-le 11286  df-sub 11478  df-neg 11479  df-div 11904  df-nn 12246  df-2 12308  df-n0 12506  df-z 12592  df-uz 12856  df-q 12966  df-rp 13010  df-xneg 13127  df-xadd 13128  df-xmul 13129  df-rest 17407  df-topgen 17428  df-psmet 21288  df-xmet 21289  df-bl 21291  df-mopn 21292  df-top 22840  df-topon 22857  df-bases 22893
This theorem is referenced by:  ressxms  24478  nrginvrcn  24653  resubmet  24762  tgioo2  24763  metdscn2  24817  divcnOLD  24828  divcn  24830  dfii3  24847  cncfcn  24874  metsscmetcld  25287  cmetss  25288  minveclem4a  25402  ftc1lem6  26020  ulmdvlem3  26383  abelth  26423  cxpcn3  26728  rlimcnp  26942  minvecolem4b  30760  minvecolem4  30762  hhsscms  31160  ftc1cnnc  37296
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