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Theorem metrest 24388
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 = (𝐢 β†Ύ (π‘Œ Γ— π‘Œ))
metrest.3 𝐽 = (MetOpenβ€˜πΆ)
metrest.4 𝐾 = (MetOpenβ€˜π·)
Assertion
Ref Expression
metrest ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝐽 β†Ύt π‘Œ) = 𝐾)
Proof of Theorem metrest
Dummy variables 𝑒 π‘Ÿ π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 4223 . . . . . . . . . 10 (𝑒 ∩ π‘Œ) βŠ† 𝑒
2 metrest.3 . . . . . . . . . . . . 13 𝐽 = (MetOpenβ€˜πΆ)
32elmopn2 24306 . . . . . . . . . . . 12 (𝐢 ∈ (∞Metβ€˜π‘‹) β†’ (𝑒 ∈ 𝐽 ↔ (𝑒 βŠ† 𝑋 ∧ βˆ€π‘¦ ∈ 𝑒 βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜πΆ)π‘Ÿ) βŠ† 𝑒)))
43simplbda 499 . . . . . . . . . . 11 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑒 ∈ 𝐽) β†’ βˆ€π‘¦ ∈ 𝑒 βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜πΆ)π‘Ÿ) βŠ† 𝑒)
54adantlr 712 . . . . . . . . . 10 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ 𝑒 ∈ 𝐽) β†’ βˆ€π‘¦ ∈ 𝑒 βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜πΆ)π‘Ÿ) βŠ† 𝑒)
6 ssralv 4045 . . . . . . . . . 10 ((𝑒 ∩ π‘Œ) βŠ† 𝑒 β†’ (βˆ€π‘¦ ∈ 𝑒 βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜πΆ)π‘Ÿ) βŠ† 𝑒 β†’ βˆ€π‘¦ ∈ (𝑒 ∩ π‘Œ)βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜πΆ)π‘Ÿ) βŠ† 𝑒))
71, 5, 6mpsyl 68 . . . . . . . . 9 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ 𝑒 ∈ 𝐽) β†’ βˆ€π‘¦ ∈ (𝑒 ∩ π‘Œ)βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜πΆ)π‘Ÿ) βŠ† 𝑒)
8 ssrin 4228 . . . . . . . . . . 11 ((𝑦(ballβ€˜πΆ)π‘Ÿ) βŠ† 𝑒 β†’ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† (𝑒 ∩ π‘Œ))
98reximi 3078 . . . . . . . . . 10 (βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜πΆ)π‘Ÿ) βŠ† 𝑒 β†’ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† (𝑒 ∩ π‘Œ))
109ralimi 3077 . . . . . . . . 9 (βˆ€π‘¦ ∈ (𝑒 ∩ π‘Œ)βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜πΆ)π‘Ÿ) βŠ† 𝑒 β†’ βˆ€π‘¦ ∈ (𝑒 ∩ π‘Œ)βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† (𝑒 ∩ π‘Œ))
117, 10syl 17 . . . . . . . 8 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ 𝑒 ∈ 𝐽) β†’ βˆ€π‘¦ ∈ (𝑒 ∩ π‘Œ)βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† (𝑒 ∩ π‘Œ))
12 inss2 4224 . . . . . . . 8 (𝑒 ∩ π‘Œ) βŠ† π‘Œ
1311, 12jctil 519 . . . . . . 7 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ 𝑒 ∈ 𝐽) β†’ ((𝑒 ∩ π‘Œ) βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ (𝑒 ∩ π‘Œ)βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† (𝑒 ∩ π‘Œ)))
14 sseq1 4002 . . . . . . . 8 (π‘₯ = (𝑒 ∩ π‘Œ) β†’ (π‘₯ βŠ† π‘Œ ↔ (𝑒 ∩ π‘Œ) βŠ† π‘Œ))
15 sseq2 4003 . . . . . . . . . 10 (π‘₯ = (𝑒 ∩ π‘Œ) β†’ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ↔ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† (𝑒 ∩ π‘Œ)))
1615rexbidv 3172 . . . . . . . . 9 (π‘₯ = (𝑒 ∩ π‘Œ) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ↔ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† (𝑒 ∩ π‘Œ)))
1716raleqbi1dv 3327 . . . . . . . 8 (π‘₯ = (𝑒 ∩ π‘Œ) β†’ (βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ↔ βˆ€π‘¦ ∈ (𝑒 ∩ π‘Œ)βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† (𝑒 ∩ π‘Œ)))
1814, 17anbi12d 630 . . . . . . 7 (π‘₯ = (𝑒 ∩ π‘Œ) β†’ ((π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯) ↔ ((𝑒 ∩ π‘Œ) βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ (𝑒 ∩ π‘Œ)βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† (𝑒 ∩ π‘Œ))))
1913, 18syl5ibrcom 246 . . . . . 6 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ 𝑒 ∈ 𝐽) β†’ (π‘₯ = (𝑒 ∩ π‘Œ) β†’ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)))
2019rexlimdva 3149 . . . . 5 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (βˆƒπ‘’ ∈ 𝐽 π‘₯ = (𝑒 ∩ π‘Œ) β†’ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)))
212mopntop 24301 . . . . . . . . 9 (𝐢 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2221ad2antrr 723 . . . . . . . 8 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)) β†’ 𝐽 ∈ Top)
23 ssel2 3972 . . . . . . . . . . . . . 14 ((π‘₯ βŠ† π‘Œ ∧ 𝑦 ∈ π‘₯) β†’ 𝑦 ∈ π‘Œ)
24 ssel2 3972 . . . . . . . . . . . . . . . 16 ((π‘Œ βŠ† 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝑦 ∈ 𝑋)
25 rpxr 12989 . . . . . . . . . . . . . . . . . 18 (π‘Ÿ ∈ ℝ+ β†’ π‘Ÿ ∈ ℝ*)
262blopn 24364 . . . . . . . . . . . . . . . . . . . 20 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑦(ballβ€˜πΆ)π‘Ÿ) ∈ 𝐽)
27 eleq1a 2822 . . . . . . . . . . . . . . . . . . . 20 ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∈ 𝐽 β†’ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
2826, 27syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
29283expa 1115 . . . . . . . . . . . . . . . . . 18 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
3025, 29sylan2 592 . . . . . . . . . . . . . . . . 17 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ π‘Ÿ ∈ ℝ+) β†’ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
3130rexlimdva 3149 . . . . . . . . . . . . . . . 16 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
3224, 31sylan2 592 . . . . . . . . . . . . . . 15 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑦 ∈ π‘Œ)) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
3332anassrs 467 . . . . . . . . . . . . . 14 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
3423, 33sylan2 592 . . . . . . . . . . . . 13 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (π‘₯ βŠ† π‘Œ ∧ 𝑦 ∈ π‘₯)) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
3534anassrs 467 . . . . . . . . . . . 12 ((((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ π‘₯ βŠ† π‘Œ) ∧ 𝑦 ∈ π‘₯) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
3635rexlimdva 3149 . . . . . . . . . . 11 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ π‘₯ βŠ† π‘Œ) β†’ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
3736adantrd 491 . . . . . . . . . 10 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ π‘₯ βŠ† π‘Œ) β†’ ((βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯) β†’ 𝑧 ∈ 𝐽))
3837adantrr 714 . . . . . . . . 9 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)) β†’ ((βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯) β†’ 𝑧 ∈ 𝐽))
3938abssdv 4060 . . . . . . . 8 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)) β†’ {𝑧 ∣ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)} βŠ† 𝐽)
40 uniopn 22754 . . . . . . . 8 ((𝐽 ∈ Top ∧ {𝑧 ∣ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)} βŠ† 𝐽) β†’ βˆͺ {𝑧 ∣ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)} ∈ 𝐽)
4122, 39, 40syl2anc 583 . . . . . . 7 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)) β†’ βˆͺ {𝑧 ∣ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)} ∈ 𝐽)
42 oveq1 7412 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑒 β†’ (𝑦(ballβ€˜πΆ)π‘Ÿ) = (𝑒(ballβ€˜πΆ)π‘Ÿ))
4342ineq1d 4206 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑒 β†’ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) = ((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ))
4443sseq1d 4008 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑒 β†’ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ↔ ((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯))
4544rexbidv 3172 . . . . . . . . . . . . . . 15 (𝑦 = 𝑒 β†’ (βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ↔ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯))
4645rspccv 3603 . . . . . . . . . . . . . 14 (βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ β†’ (𝑒 ∈ π‘₯ β†’ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯))
4746ad2antll 726 . . . . . . . . . . . . 13 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)) β†’ (𝑒 ∈ π‘₯ β†’ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯))
48 ssel 3970 . . . . . . . . . . . . . . 15 (π‘₯ βŠ† π‘Œ β†’ (𝑒 ∈ π‘₯ β†’ 𝑒 ∈ π‘Œ))
49 ssel 3970 . . . . . . . . . . . . . . . 16 (π‘Œ βŠ† 𝑋 β†’ (𝑒 ∈ π‘Œ β†’ 𝑒 ∈ 𝑋))
50 blcntr 24274 . . . . . . . . . . . . . . . . . . . . 21 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑒 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) β†’ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ))
5150a1d 25 . . . . . . . . . . . . . . . . . . . 20 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑒 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) β†’ (((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ β†’ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ)))
5251ancld 550 . . . . . . . . . . . . . . . . . . 19 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑒 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) β†’ (((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ β†’ (((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ))))
53523expa 1115 . . . . . . . . . . . . . . . . . 18 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑒 ∈ 𝑋) ∧ π‘Ÿ ∈ ℝ+) β†’ (((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ β†’ (((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ))))
5453reximdva 3162 . . . . . . . . . . . . . . . . 17 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑒 ∈ 𝑋) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ ((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ β†’ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ))))
5554ex 412 . . . . . . . . . . . . . . . 16 (𝐢 ∈ (∞Metβ€˜π‘‹) β†’ (𝑒 ∈ 𝑋 β†’ (βˆƒπ‘Ÿ ∈ ℝ+ ((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ β†’ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ)))))
5649, 55sylan9r 508 . . . . . . . . . . . . . . 15 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝑒 ∈ π‘Œ β†’ (βˆƒπ‘Ÿ ∈ ℝ+ ((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ β†’ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ)))))
5748, 56sylan9r 508 . . . . . . . . . . . . . 14 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ π‘₯ βŠ† π‘Œ) β†’ (𝑒 ∈ π‘₯ β†’ (βˆƒπ‘Ÿ ∈ ℝ+ ((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ β†’ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ)))))
5857adantrr 714 . . . . . . . . . . . . 13 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)) β†’ (𝑒 ∈ π‘₯ β†’ (βˆƒπ‘Ÿ ∈ ℝ+ ((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ β†’ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ)))))
5947, 58mpdd 43 . . . . . . . . . . . 12 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)) β†’ (𝑒 ∈ π‘₯ β†’ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ))))
6042eleq2d 2813 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑒 β†’ (𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ) ↔ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ)))
6144, 60anbi12d 630 . . . . . . . . . . . . . . 15 (𝑦 = 𝑒 β†’ ((((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)) ↔ (((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ))))
6261rexbidv 3172 . . . . . . . . . . . . . 14 (𝑦 = 𝑒 β†’ (βˆƒπ‘Ÿ ∈ ℝ+ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)) ↔ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ))))
6362rspcev 3606 . . . . . . . . . . . . 13 ((𝑒 ∈ π‘₯ ∧ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ))) β†’ βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)))
6463ex 412 . . . . . . . . . . . 12 (𝑒 ∈ π‘₯ β†’ (βˆƒπ‘Ÿ ∈ ℝ+ (((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ)) β†’ βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ))))
6559, 64sylcom 30 . . . . . . . . . . 11 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)) β†’ (𝑒 ∈ π‘₯ β†’ βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ))))
66 simprl 768 . . . . . . . . . . . 12 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)) β†’ π‘₯ βŠ† π‘Œ)
6766sseld 3976 . . . . . . . . . . 11 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)) β†’ (𝑒 ∈ π‘₯ β†’ 𝑒 ∈ π‘Œ))
6865, 67jcad 512 . . . . . . . . . 10 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)) β†’ (𝑒 ∈ π‘₯ β†’ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)) ∧ 𝑒 ∈ π‘Œ)))
69 elin 3959 . . . . . . . . . . . . . . 15 (𝑒 ∈ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) ↔ (𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ 𝑒 ∈ π‘Œ))
70 ssel2 3972 . . . . . . . . . . . . . . 15 ((((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ)) β†’ 𝑒 ∈ π‘₯)
7169, 70sylan2br 594 . . . . . . . . . . . . . 14 ((((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ (𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ 𝑒 ∈ π‘Œ)) β†’ 𝑒 ∈ π‘₯)
7271expr 456 . . . . . . . . . . . . 13 ((((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)) β†’ (𝑒 ∈ π‘Œ β†’ 𝑒 ∈ π‘₯))
7372rexlimivw 3145 . . . . . . . . . . . 12 (βˆƒπ‘Ÿ ∈ ℝ+ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)) β†’ (𝑒 ∈ π‘Œ β†’ 𝑒 ∈ π‘₯))
7473rexlimivw 3145 . . . . . . . . . . 11 (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)) β†’ (𝑒 ∈ π‘Œ β†’ 𝑒 ∈ π‘₯))
7574imp 406 . . . . . . . . . 10 ((βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)) ∧ 𝑒 ∈ π‘Œ) β†’ 𝑒 ∈ π‘₯)
7668, 75impbid1 224 . . . . . . . . 9 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)) β†’ (𝑒 ∈ π‘₯ ↔ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)) ∧ 𝑒 ∈ π‘Œ)))
77 elin 3959 . . . . . . . . . 10 (𝑒 ∈ (βˆͺ {𝑧 ∣ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)} ∩ π‘Œ) ↔ (𝑒 ∈ βˆͺ {𝑧 ∣ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)} ∧ 𝑒 ∈ π‘Œ))
78 eluniab 4916 . . . . . . . . . . . 12 (𝑒 ∈ βˆͺ {𝑧 ∣ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)} ↔ βˆƒπ‘§(𝑒 ∈ 𝑧 ∧ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)))
79 ancom 460 . . . . . . . . . . . . . 14 ((𝑒 ∈ 𝑧 ∧ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)) ↔ ((βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯) ∧ 𝑒 ∈ 𝑧))
80 anass 468 . . . . . . . . . . . . . 14 (((βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯) ∧ 𝑒 ∈ 𝑧) ↔ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)))
81 r19.41v 3182 . . . . . . . . . . . . . . . 16 (βˆƒπ‘Ÿ ∈ ℝ+ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)) ↔ (βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)))
8281rexbii 3088 . . . . . . . . . . . . . . 15 (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)) ↔ βˆƒπ‘¦ ∈ π‘₯ (βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)))
83 r19.41v 3182 . . . . . . . . . . . . . . 15 (βˆƒπ‘¦ ∈ π‘₯ (βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)) ↔ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)))
8482, 83bitr2i 276 . . . . . . . . . . . . . 14 ((βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)) ↔ βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)))
8579, 80, 843bitri 297 . . . . . . . . . . . . 13 ((𝑒 ∈ 𝑧 ∧ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)) ↔ βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)))
8685exbii 1842 . . . . . . . . . . . 12 (βˆƒπ‘§(𝑒 ∈ 𝑧 ∧ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)) ↔ βˆƒπ‘§βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)))
87 ovex 7438 . . . . . . . . . . . . . . . . 17 (𝑦(ballβ€˜πΆ)π‘Ÿ) ∈ V
88 ineq1 4200 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ (𝑧 ∩ π‘Œ) = ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ))
8988sseq1d 4008 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ↔ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯))
90 eleq2 2816 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ (𝑒 ∈ 𝑧 ↔ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)))
9189, 90anbi12d 630 . . . . . . . . . . . . . . . . 17 (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ (((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧) ↔ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ))))
9287, 91ceqsexv 3520 . . . . . . . . . . . . . . . 16 (βˆƒπ‘§(𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)) ↔ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)))
9392rexbii 3088 . . . . . . . . . . . . . . 15 (βˆƒπ‘Ÿ ∈ ℝ+ βˆƒπ‘§(𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)) ↔ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)))
94 rexcom4 3279 . . . . . . . . . . . . . . 15 (βˆƒπ‘Ÿ ∈ ℝ+ βˆƒπ‘§(𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)) ↔ βˆƒπ‘§βˆƒπ‘Ÿ ∈ ℝ+ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)))
9593, 94bitr3i 277 . . . . . . . . . . . . . 14 (βˆƒπ‘Ÿ ∈ ℝ+ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)) ↔ βˆƒπ‘§βˆƒπ‘Ÿ ∈ ℝ+ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)))
9695rexbii 3088 . . . . . . . . . . . . 13 (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)) ↔ βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘§βˆƒπ‘Ÿ ∈ ℝ+ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)))
97 rexcom4 3279 . . . . . . . . . . . . 13 (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘§βˆƒπ‘Ÿ ∈ ℝ+ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)) ↔ βˆƒπ‘§βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)))
9896, 97bitr2i 276 . . . . . . . . . . . 12 (βˆƒπ‘§βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ 𝑧)) ↔ βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)))
9978, 86, 983bitri 297 . . . . . . . . . . 11 (𝑒 ∈ βˆͺ {𝑧 ∣ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)} ↔ βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)))
10099anbi1i 623 . . . . . . . . . 10 ((𝑒 ∈ βˆͺ {𝑧 ∣ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)} ∧ 𝑒 ∈ π‘Œ) ↔ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)) ∧ 𝑒 ∈ π‘Œ))
10177, 100bitr2i 276 . . . . . . . . 9 ((βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯ ∧ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)) ∧ 𝑒 ∈ π‘Œ) ↔ 𝑒 ∈ (βˆͺ {𝑧 ∣ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)} ∩ π‘Œ))
10276, 101bitrdi 287 . . . . . . . 8 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)) β†’ (𝑒 ∈ π‘₯ ↔ 𝑒 ∈ (βˆͺ {𝑧 ∣ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)} ∩ π‘Œ)))
103102eqrdv 2724 . . . . . . 7 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)) β†’ π‘₯ = (βˆͺ {𝑧 ∣ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)} ∩ π‘Œ))
104 ineq1 4200 . . . . . . . 8 (𝑒 = βˆͺ {𝑧 ∣ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)} β†’ (𝑒 ∩ π‘Œ) = (βˆͺ {𝑧 ∣ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)} ∩ π‘Œ))
105104rspceeqv 3628 . . . . . . 7 ((βˆͺ {𝑧 ∣ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)} ∈ 𝐽 ∧ π‘₯ = (βˆͺ {𝑧 ∣ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)} ∩ π‘Œ)) β†’ βˆƒπ‘’ ∈ 𝐽 π‘₯ = (𝑒 ∩ π‘Œ))
10641, 103, 105syl2anc 583 . . . . . 6 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)) β†’ βˆƒπ‘’ ∈ 𝐽 π‘₯ = (𝑒 ∩ π‘Œ))
107106ex 412 . . . . 5 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ ((π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯) β†’ βˆƒπ‘’ ∈ 𝐽 π‘₯ = (𝑒 ∩ π‘Œ)))
10820, 107impbid 211 . . . 4 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (βˆƒπ‘’ ∈ 𝐽 π‘₯ = (𝑒 ∩ π‘Œ) ↔ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)))
109 simpr 484 . . . . . . . . . . 11 ((π‘Œ βŠ† 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝑦 ∈ π‘Œ)
11024, 109elind 4189 . . . . . . . . . 10 ((π‘Œ βŠ† 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝑦 ∈ (𝑋 ∩ π‘Œ))
111 metrest.1 . . . . . . . . . . . . . . 15 𝐷 = (𝐢 β†Ύ (π‘Œ Γ— π‘Œ))
112111blres 24292 . . . . . . . . . . . . . 14 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ (𝑋 ∩ π‘Œ) ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑦(ballβ€˜π·)π‘Ÿ) = ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ))
113112sseq1d 4008 . . . . . . . . . . . . 13 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ (𝑋 ∩ π‘Œ) ∧ π‘Ÿ ∈ ℝ*) β†’ ((𝑦(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ ↔ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯))
1141133expa 1115 . . . . . . . . . . . 12 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ (𝑋 ∩ π‘Œ)) ∧ π‘Ÿ ∈ ℝ*) β†’ ((𝑦(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ ↔ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯))
11525, 114sylan2 592 . . . . . . . . . . 11 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ (𝑋 ∩ π‘Œ)) ∧ π‘Ÿ ∈ ℝ+) β†’ ((𝑦(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ ↔ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯))
116115rexbidva 3170 . . . . . . . . . 10 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ (𝑋 ∩ π‘Œ)) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ ↔ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯))
117110, 116sylan2 592 . . . . . . . . 9 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ (π‘Œ βŠ† 𝑋 ∧ 𝑦 ∈ π‘Œ)) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ ↔ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯))
118117anassrs 467 . . . . . . . 8 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ 𝑦 ∈ π‘Œ) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ ↔ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯))
11923, 118sylan2 592 . . . . . . 7 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ (π‘₯ βŠ† π‘Œ ∧ 𝑦 ∈ π‘₯)) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ ↔ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯))
120119anassrs 467 . . . . . 6 ((((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ π‘₯ βŠ† π‘Œ) ∧ 𝑦 ∈ π‘₯) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ ↔ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯))
121120ralbidva 3169 . . . . 5 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) ∧ π‘₯ βŠ† π‘Œ) β†’ (βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯ ↔ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯))
122121pm5.32da 578 . . . 4 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ ((π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯) ↔ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯)))
123108, 122bitr4d 282 . . 3 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (βˆƒπ‘’ ∈ 𝐽 π‘₯ = (𝑒 ∩ π‘Œ) ↔ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯)))
124 id 22 . . . . 5 (π‘Œ βŠ† 𝑋 β†’ π‘Œ βŠ† 𝑋)
1252mopnm 24305 . . . . 5 (𝐢 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
126 ssexg 5316 . . . . 5 ((π‘Œ βŠ† 𝑋 ∧ 𝑋 ∈ 𝐽) β†’ π‘Œ ∈ V)
127124, 125, 126syl2anr 596 . . . 4 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ π‘Œ ∈ V)
128 elrest 17382 . . . 4 ((𝐽 ∈ Top ∧ π‘Œ ∈ V) β†’ (π‘₯ ∈ (𝐽 β†Ύt π‘Œ) ↔ βˆƒπ‘’ ∈ 𝐽 π‘₯ = (𝑒 ∩ π‘Œ)))
12921, 127, 128syl2an2r 682 . . 3 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (π‘₯ ∈ (𝐽 β†Ύt π‘Œ) ↔ βˆƒπ‘’ ∈ 𝐽 π‘₯ = (𝑒 ∩ π‘Œ)))
130 xmetres2 24222 . . . . 5 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝐢 β†Ύ (π‘Œ Γ— π‘Œ)) ∈ (∞Metβ€˜π‘Œ))
131111, 130eqeltrid 2831 . . . 4 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ 𝐷 ∈ (∞Metβ€˜π‘Œ))
132 metrest.4 . . . . 5 𝐾 = (MetOpenβ€˜π·)
133132elmopn2 24306 . . . 4 (𝐷 ∈ (∞Metβ€˜π‘Œ) β†’ (π‘₯ ∈ 𝐾 ↔ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯)))
134131, 133syl 17 . . 3 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (π‘₯ ∈ 𝐾 ↔ (π‘₯ βŠ† π‘Œ ∧ βˆ€π‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜π·)π‘Ÿ) βŠ† π‘₯)))
135123, 129, 1343bitr4d 311 . 2 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (π‘₯ ∈ (𝐽 β†Ύt π‘Œ) ↔ π‘₯ ∈ 𝐾))
136135eqrdv 2724 1 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝐽 β†Ύt π‘Œ) = 𝐾)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533  βˆƒwex 1773   ∈ wcel 2098  {cab 2703  βˆ€wral 3055  βˆƒwrex 3064  Vcvv 3468   ∩ cin 3942   βŠ† wss 3943  βˆͺ cuni 4902   Γ— cxp 5667   β†Ύ cres 5671  β€˜cfv 6537  (class class class)co 7405  β„*cxr 11251  β„+crp 12980   β†Ύt crest 17375  βˆžMetcxmet 21225  ballcbl 21227  MetOpencmopn 21230  Topctop 22750
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 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-sup 9439  df-inf 9440  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-n0 12477  df-z 12563  df-uz 12827  df-q 12937  df-rp 12981  df-xneg 13098  df-xadd 13099  df-xmul 13100  df-rest 17377  df-topgen 17398  df-psmet 21232  df-xmet 21233  df-bl 21235  df-mopn 21236  df-top 22751  df-topon 22768  df-bases 22804
This theorem is referenced by:  ressxms  24389  nrginvrcn  24564  resubmet  24673  tgioo2  24674  metdscn2  24728  divcnOLD  24739  divcn  24741  dfii3  24758  cncfcn  24785  metsscmetcld  25198  cmetss  25199  minveclem4a  25313  ftc1lem6  25931  ulmdvlem3  26293  abelth  26333  cxpcn3  26638  rlimcnp  26852  minvecolem4b  30640  minvecolem4  30642  hhsscms  31040  ftc1cnnc  37073
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