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Theorem metrest 23785
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 4179 . . . . . . . . . 10 (𝑢𝑌) ⊆ 𝑢
2 metrest.3 . . . . . . . . . . . . 13 𝐽 = (MetOpen‘𝐶)
32elmopn2 23703 . . . . . . . . . . . 12 (𝐶 ∈ (∞Met‘𝑋) → (𝑢𝐽 ↔ (𝑢𝑋 ∧ ∀𝑦𝑢𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢)))
43simplbda 501 . . . . . . . . . . 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 4001 . . . . . . . . . 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 4184 . . . . . . . . . . 11 ((𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌))
98reximi 3084 . . . . . . . . . 10 (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐶)𝑟) ⊆ 𝑢 → ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌))
109ralimi 3083 . . . . . . . . 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 4180 . . . . . . . 8 (𝑢𝑌) ⊆ 𝑌
1311, 12jctil 521 . . . . . . 7 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑢𝐽) → ((𝑢𝑌) ⊆ 𝑌 ∧ ∀𝑦 ∈ (𝑢𝑌)∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌)))
14 sseq1 3960 . . . . . . . 8 (𝑥 = (𝑢𝑌) → (𝑥𝑌 ↔ (𝑢𝑌) ⊆ 𝑌))
15 sseq2 3961 . . . . . . . . . 10 (𝑥 = (𝑢𝑌) → (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌)))
1615rexbidv 3172 . . . . . . . . 9 (𝑥 = (𝑢𝑌) → (∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ (𝑢𝑌)))
1716raleqbi1dv 3304 . . . . . . . 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 3149 . . . . 5 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (∃𝑢𝐽 𝑥 = (𝑢𝑌) → (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)))
212mopntop 23698 . . . . . . . . 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 3930 . . . . . . . . . . . . . 14 ((𝑥𝑌𝑦𝑥) → 𝑦𝑌)
24 ssel2 3930 . . . . . . . . . . . . . . . 16 ((𝑌𝑋𝑦𝑌) → 𝑦𝑋)
25 rpxr 12844 . . . . . . . . . . . . . . . . . 18 (𝑟 ∈ ℝ+𝑟 ∈ ℝ*)
262blopn 23761 . . . . . . . . . . . . . . . . . . . 20 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋𝑟 ∈ ℝ*) → (𝑦(ball‘𝐶)𝑟) ∈ 𝐽)
27 eleq1a 2833 . . . . . . . . . . . . . . . . . . . 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 594 . . . . . . . . . . . . . . . . 17 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋) ∧ 𝑟 ∈ ℝ+) → (𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3130rexlimdva 3149 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦𝑋) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3224, 31sylan2 594 . . . . . . . . . . . . . . 15 ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑌𝑋𝑦𝑌)) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3332anassrs 469 . . . . . . . . . . . . . 14 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑦𝑌) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3423, 33sylan2 594 . . . . . . . . . . . . 13 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌𝑦𝑥)) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3534anassrs 469 . . . . . . . . . . . 12 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) ∧ 𝑦𝑥) → (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3635rexlimdva 3149 . . . . . . . . . . 11 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) → (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) → 𝑧𝐽))
3736adantrd 493 . . . . . . . . . 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 4016 . . . . . . . 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 22151 . . . . . . . 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 585 . . . . . . 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 7348 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑢 → (𝑦(ball‘𝐶)𝑟) = (𝑢(ball‘𝐶)𝑟))
4342ineq1d 4162 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑢 → ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) = ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌))
4443sseq1d 3966 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑢 → (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
4544rexbidv 3172 . . . . . . . . . . . . . . 15 (𝑦 = 𝑢 → (∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑢(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
4645rspccv 3570 . . . . . . . . . . . . . 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 3928 . . . . . . . . . . . . . . 15 (𝑥𝑌 → (𝑢𝑥𝑢𝑌))
49 ssel 3928 . . . . . . . . . . . . . . . 16 (𝑌𝑋 → (𝑢𝑌𝑢𝑋))
50 blcntr 23671 . . . . . . . . . . . . . . . . . . . . 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 552 . . . . . . . . . . . . . . . . . . 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 3162 . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . . 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 510 . . . . . . . . . . . . . . 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 510 . . . . . . . . . . . . . 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 2823 . . . . . . . . . . . . . . . 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 3172 . . . . . . . . . . . . . 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 3573 . . . . . . . . . . . . 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 414 . . . . . . . . . . . 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 3934 . . . . . . . . . . 11 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → (𝑢𝑥𝑢𝑌))
6865, 67jcad 514 . . . . . . . . . 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 3917 . . . . . . . . . . . . . . 15 (𝑢 ∈ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ↔ (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ∧ 𝑢𝑌))
70 ssel2 3930 . . . . . . . . . . . . . . 15 ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌)) → 𝑢𝑥)
7169, 70sylan2br 596 . . . . . . . . . . . . . 14 ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥 ∧ (𝑢 ∈ (𝑦(ball‘𝐶)𝑟) ∧ 𝑢𝑌)) → 𝑢𝑥)
7271expr 458 . . . . . . . . . . . . 13 ((((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢𝑌𝑢𝑥))
7372rexlimivw 3145 . . . . . . . . . . . 12 (∃𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢𝑌𝑢𝑥))
7473rexlimivw 3145 . . . . . . . . . . 11 (∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) → (𝑢𝑌𝑢𝑥))
7574imp 408 . . . . . . . . . 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 3917 . . . . . . . . . 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 4871 . . . . . . . . . . . 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 462 . . . . . . . . . . . . . 14 ((𝑢𝑧 ∧ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)) ↔ ((∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥) ∧ 𝑢𝑧))
80 anass 470 . . . . . . . . . . . . . 14 (((∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥) ∧ 𝑢𝑧) ↔ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
81 r19.41v 3182 . . . . . . . . . . . . . . . 16 (∃𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
8281rexbii 3094 . . . . . . . . . . . . . . 15 (∃𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ ∃𝑦𝑥 (∃𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
83 r19.41v 3182 . . . . . . . . . . . . . . 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 1850 . . . . . . . . . . . 12 (∃𝑧(𝑢𝑧 ∧ (∃𝑦𝑥𝑟 ∈ ℝ+ 𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ (𝑧𝑌) ⊆ 𝑥)) ↔ ∃𝑧𝑦𝑥𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
87 ovex 7374 . . . . . . . . . . . . . . . . 17 (𝑦(ball‘𝐶)𝑟) ∈ V
88 ineq1 4156 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑦(ball‘𝐶)𝑟) → (𝑧𝑌) = ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌))
8988sseq1d 3966 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑦(ball‘𝐶)𝑟) → ((𝑧𝑌) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
90 eleq2 2826 . . . . . . . . . . . . . . . . . 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 3490 . . . . . . . . . . . . . . . 16 (∃𝑧(𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)) ↔ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)))
9392rexbii 3094 . . . . . . . . . . . . . . 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 3268 . . . . . . . . . . . . . . 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 3094 . . . . . . . . . . . . 13 (∃𝑦𝑥𝑟 ∈ ℝ+ (((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥𝑢 ∈ (𝑦(ball‘𝐶)𝑟)) ↔ ∃𝑦𝑥𝑧𝑟 ∈ ℝ+ (𝑧 = (𝑦(ball‘𝐶)𝑟) ∧ ((𝑧𝑌) ⊆ 𝑥𝑢𝑧)))
97 rexcom4 3268 . . . . . . . . . . . . 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 625 . . . . . . . . . 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 2735 . . . . . . 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 4156 . . . . . . . 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 3587 . . . . . . 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 585 . . . . . 6 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌 ∧ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥)) → ∃𝑢𝐽 𝑥 = (𝑢𝑌))
107106ex 414 . . . . 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 486 . . . . . . . . . . 11 ((𝑌𝑋𝑦𝑌) → 𝑦𝑌)
11024, 109elind 4145 . . . . . . . . . 10 ((𝑌𝑋𝑦𝑌) → 𝑦 ∈ (𝑋𝑌))
111 metrest.1 . . . . . . . . . . . . . . 15 𝐷 = (𝐶 ↾ (𝑌 × 𝑌))
112111blres 23689 . . . . . . . . . . . . . 14 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌) ∧ 𝑟 ∈ ℝ*) → (𝑦(ball‘𝐷)𝑟) = ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌))
113112sseq1d 3966 . . . . . . . . . . . . 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 594 . . . . . . . . . . 11 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌)) ∧ 𝑟 ∈ ℝ+) → ((𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
116115rexbidva 3170 . . . . . . . . . 10 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ (𝑋𝑌)) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
117110, 116sylan2 594 . . . . . . . . 9 ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑌𝑋𝑦𝑌)) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
118117anassrs 469 . . . . . . . 8 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑦𝑌) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
11923, 118sylan2 594 . . . . . . 7 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ (𝑥𝑌𝑦𝑥)) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
120119anassrs 469 . . . . . 6 ((((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) ∧ 𝑦𝑥) → (∃𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∃𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
121120ralbidva 3169 . . . . 5 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑥𝑌) → (∀𝑦𝑥𝑟 ∈ ℝ+ (𝑦(ball‘𝐷)𝑟) ⊆ 𝑥 ↔ ∀𝑦𝑥𝑟 ∈ ℝ+ ((𝑦(ball‘𝐶)𝑟) ∩ 𝑌) ⊆ 𝑥))
122121pm5.32da 580 . . . 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 23702 . . . . 5 (𝐶 ∈ (∞Met‘𝑋) → 𝑋𝐽)
126 ssexg 5271 . . . . 5 ((𝑌𝑋𝑋𝐽) → 𝑌 ∈ V)
127124, 125, 126syl2anr 598 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → 𝑌 ∈ V)
128 elrest 17235 . . . 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 23619 . . . . 5 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝐶 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌))
131111, 130eqeltrid 2842 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → 𝐷 ∈ (∞Met‘𝑌))
132 metrest.4 . . . . 5 𝐾 = (MetOpen‘𝐷)
133132elmopn2 23703 . . . 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 2735 1 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑌𝑋) → (𝐽t 𝑌) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2714  wral 3062  wrex 3071  Vcvv 3442  cin 3900  wss 3901   cuni 4856   × cxp 5622  cres 5626  cfv 6483  (class class class)co 7341  *cxr 11113  +crp 12835  t crest 17228  ∞Metcxmet 20687  ballcbl 20689  MetOpencmopn 20692  Topctop 22147
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5233  ax-sep 5247  ax-nul 5254  ax-pow 5312  ax-pr 5376  ax-un 7654  ax-cnex 11032  ax-resscn 11033  ax-1cn 11034  ax-icn 11035  ax-addcl 11036  ax-addrcl 11037  ax-mulcl 11038  ax-mulrcl 11039  ax-mulcom 11040  ax-addass 11041  ax-mulass 11042  ax-distr 11043  ax-i2m1 11044  ax-1ne0 11045  ax-1rid 11046  ax-rnegex 11047  ax-rrecex 11048  ax-cnre 11049  ax-pre-lttri 11050  ax-pre-lttrn 11051  ax-pre-ltadd 11052  ax-pre-mulgt0 11053  ax-pre-sup 11054
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3350  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3731  df-csb 3847  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-pss 3920  df-nul 4274  df-if 4478  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-iun 4947  df-br 5097  df-opab 5159  df-mpt 5180  df-tr 5214  df-id 5522  df-eprel 5528  df-po 5536  df-so 5537  df-fr 5579  df-we 5581  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6242  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6435  df-fun 6485  df-fn 6486  df-f 6487  df-f1 6488  df-fo 6489  df-f1o 6490  df-fv 6491  df-riota 7297  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7785  df-1st 7903  df-2nd 7904  df-frecs 8171  df-wrecs 8202  df-recs 8276  df-rdg 8315  df-er 8573  df-map 8692  df-en 8809  df-dom 8810  df-sdom 8811  df-sup 9303  df-inf 9304  df-pnf 11116  df-mnf 11117  df-xr 11118  df-ltxr 11119  df-le 11120  df-sub 11312  df-neg 11313  df-div 11738  df-nn 12079  df-2 12141  df-n0 12339  df-z 12425  df-uz 12688  df-q 12794  df-rp 12836  df-xneg 12953  df-xadd 12954  df-xmul 12955  df-rest 17230  df-topgen 17251  df-psmet 20694  df-xmet 20695  df-bl 20697  df-mopn 20698  df-top 22148  df-topon 22165  df-bases 22201
This theorem is referenced by:  ressxms  23786  nrginvrcn  23961  resubmet  24070  tgioo2  24071  metdscn2  24125  divcn  24136  dfii3  24151  cncfcn  24178  metsscmetcld  24584  cmetss  24585  minveclem4a  24699  ftc1lem6  25310  ulmdvlem3  25666  abelth  25705  cxpcn3  26006  rlimcnp  26220  minvecolem4b  29527  minvecolem4  29529  hhsscms  29927  ftc1cnnc  36005
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