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Theorem metrest 23896
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 4193 . . . . . . . . . 10 (𝑒 ∩ π‘Œ) βŠ† 𝑒
2 metrest.3 . . . . . . . . . . . . 13 𝐽 = (MetOpenβ€˜πΆ)
32elmopn2 23814 . . . . . . . . . . . 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 501 . . . . . . . . . . 11 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑒 ∈ 𝐽) β†’ βˆ€π‘¦ ∈ 𝑒 βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜πΆ)π‘Ÿ) βŠ† 𝑒)
54adantlr 714 . . . . . . . . . 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 4015 . . . . . . . . . 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 4198 . . . . . . . . . . 11 ((𝑦(ballβ€˜πΆ)π‘Ÿ) βŠ† 𝑒 β†’ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† (𝑒 ∩ π‘Œ))
98reximi 3088 . . . . . . . . . 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 3087 . . . . . . . . 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 4194 . . . . . . . 8 (𝑒 ∩ π‘Œ) βŠ† π‘Œ
1311, 12jctil 521 . . . . . . 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 3974 . . . . . . . 8 (π‘₯ = (𝑒 ∩ π‘Œ) β†’ (π‘₯ βŠ† π‘Œ ↔ (𝑒 ∩ π‘Œ) βŠ† π‘Œ))
15 sseq2 3975 . . . . . . . . . 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 3176 . . . . . . . . 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 3310 . . . . . . . 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 632 . . . . . . 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 247 . . . . . 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 3153 . . . . 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 23809 . . . . . . . . 9 (𝐢 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2221ad2antrr 725 . . . . . . . 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 3944 . . . . . . . . . . . . . 14 ((π‘₯ βŠ† π‘Œ ∧ 𝑦 ∈ π‘₯) β†’ 𝑦 ∈ π‘Œ)
24 ssel2 3944 . . . . . . . . . . . . . . . 16 ((π‘Œ βŠ† 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝑦 ∈ 𝑋)
25 rpxr 12931 . . . . . . . . . . . . . . . . . 18 (π‘Ÿ ∈ ℝ+ β†’ π‘Ÿ ∈ ℝ*)
262blopn 23872 . . . . . . . . . . . . . . . . . . . 20 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑦(ballβ€˜πΆ)π‘Ÿ) ∈ 𝐽)
27 eleq1a 2833 . . . . . . . . . . . . . . . . . . . 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 1119 . . . . . . . . . . . . . . . . . 18 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
3025, 29sylan2 594 . . . . . . . . . . . . . . . . 17 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ π‘Ÿ ∈ ℝ+) β†’ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
3130rexlimdva 3153 . . . . . . . . . . . . . . . 16 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
3224, 31sylan2 594 . . . . . . . . . . . . . . 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 469 . . . . . . . . . . . . . 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 594 . . . . . . . . . . . . 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 469 . . . . . . . . . . . 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 3153 . . . . . . . . . . 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 493 . . . . . . . . . 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 716 . . . . . . . . 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 4030 . . . . . . . 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 22262 . . . . . . . 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 585 . . . . . . 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 7369 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑒 β†’ (𝑦(ballβ€˜πΆ)π‘Ÿ) = (𝑒(ballβ€˜πΆ)π‘Ÿ))
4342ineq1d 4176 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑒 β†’ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) = ((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ))
4443sseq1d 3980 . . . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . . . . 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 3581 . . . . . . . . . . . . . 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 728 . . . . . . . . . . . . 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 3942 . . . . . . . . . . . . . . 15 (π‘₯ βŠ† π‘Œ β†’ (𝑒 ∈ π‘₯ β†’ 𝑒 ∈ π‘Œ))
49 ssel 3942 . . . . . . . . . . . . . . . 16 (π‘Œ βŠ† 𝑋 β†’ (𝑒 ∈ π‘Œ β†’ 𝑒 ∈ 𝑋))
50 blcntr 23782 . . . . . . . . . . . . . . . . . . . . 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 552 . . . . . . . . . . . . . . . . . . 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 1119 . . . . . . . . . . . . . . . . . 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 3166 . . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . . . . 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 510 . . . . . . . . . . . . . . 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 510 . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑒 β†’ (𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ) ↔ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ)))
6144, 60anbi12d 632 . . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . . . 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 3584 . . . . . . . . . . . . 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 414 . . . . . . . . . . . 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 770 . . . . . . . . . . . 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 3948 . . . . . . . . . . 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 514 . . . . . . . . . 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 3931 . . . . . . . . . . . . . . 15 (𝑒 ∈ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) ↔ (𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ 𝑒 ∈ π‘Œ))
70 ssel2 3944 . . . . . . . . . . . . . . 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 596 . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . 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 3149 . . . . . . . . . . . 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 3149 . . . . . . . . . . 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 408 . . . . . . . . . 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 3931 . . . . . . . . . 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 4885 . . . . . . . . . . . 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 462 . . . . . . . . . . . . . 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 470 . . . . . . . . . . . . . 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 3186 . . . . . . . . . . . . . . . 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 3098 . . . . . . . . . . . . . . 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 3186 . . . . . . . . . . . . . . 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 1851 . . . . . . . . . . . 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 7395 . . . . . . . . . . . . . . . . 17 (𝑦(ballβ€˜πΆ)π‘Ÿ) ∈ V
88 ineq1 4170 . . . . . . . . . . . . . . . . . . 19 (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ (𝑧 ∩ π‘Œ) = ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ))
8988sseq1d 3980 . . . . . . . . . . . . . . . . . 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 2827 . . . . . . . . . . . . . . . . . 18 (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ (𝑒 ∈ 𝑧 ↔ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)))
9189, 90anbi12d 632 . . . . . . . . . . . . . . . . 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 3497 . . . . . . . . . . . . . . . 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 3098 . . . . . . . . . . . . . . 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 3274 . . . . . . . . . . . . . . 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 3098 . . . . . . . . . . . . 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 3274 . . . . . . . . . . . . 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 625 . . . . . . . . . 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 2735 . . . . . . 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 4170 . . . . . . . 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 3600 . . . . . . 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 585 . . . . . 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 414 . . . . 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 486 . . . . . . . . . . 11 ((π‘Œ βŠ† 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝑦 ∈ π‘Œ)
11024, 109elind 4159 . . . . . . . . . 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 23800 . . . . . . . . . . . . . 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 3980 . . . . . . . . . . . . 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 1119 . . . . . . . . . . . 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 594 . . . . . . . . . . 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 3174 . . . . . . . . . 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 594 . . . . . . . . 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 469 . . . . . . . 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 594 . . . . . . 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 469 . . . . . 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 3173 . . . . 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 580 . . . 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 23813 . . . . 5 (𝐢 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
126 ssexg 5285 . . . . 5 ((π‘Œ βŠ† 𝑋 ∧ 𝑋 ∈ 𝐽) β†’ π‘Œ ∈ V)
127124, 125, 126syl2anr 598 . . . 4 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ π‘Œ ∈ V)
128 elrest 17316 . . . 4 ((𝐽 ∈ Top ∧ π‘Œ ∈ V) β†’ (π‘₯ ∈ (𝐽 β†Ύt π‘Œ) ↔ βˆƒπ‘’ ∈ 𝐽 π‘₯ = (𝑒 ∩ π‘Œ)))
12921, 127, 128syl2an2r 684 . . 3 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (π‘₯ ∈ (𝐽 β†Ύt π‘Œ) ↔ βˆƒπ‘’ ∈ 𝐽 π‘₯ = (𝑒 ∩ π‘Œ)))
130 xmetres2 23730 . . . . 5 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝐢 β†Ύ (π‘Œ Γ— π‘Œ)) ∈ (∞Metβ€˜π‘Œ))
131111, 130eqeltrid 2842 . . . 4 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ 𝐷 ∈ (∞Metβ€˜π‘Œ))
132 metrest.4 . . . . 5 𝐾 = (MetOpenβ€˜π·)
133132elmopn2 23814 . . . 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 2735 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 397   ∧ w3a 1088   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2714  βˆ€wral 3065  βˆƒwrex 3074  Vcvv 3448   ∩ cin 3914   βŠ† wss 3915  βˆͺ cuni 4870   Γ— cxp 5636   β†Ύ cres 5640  β€˜cfv 6501  (class class class)co 7362  β„*cxr 11195  β„+crp 12922   β†Ύt crest 17309  βˆžMetcxmet 20797  ballcbl 20799  MetOpencmopn 20802  Topctop 22258
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11114  ax-resscn 11115  ax-1cn 11116  ax-icn 11117  ax-addcl 11118  ax-addrcl 11119  ax-mulcl 11120  ax-mulrcl 11121  ax-mulcom 11122  ax-addass 11123  ax-mulass 11124  ax-distr 11125  ax-i2m1 11126  ax-1ne0 11127  ax-1rid 11128  ax-rnegex 11129  ax-rrecex 11130  ax-cnre 11131  ax-pre-lttri 11132  ax-pre-lttrn 11133  ax-pre-ltadd 11134  ax-pre-mulgt0 11135  ax-pre-sup 11136
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-er 8655  df-map 8774  df-en 8891  df-dom 8892  df-sdom 8893  df-sup 9385  df-inf 9386  df-pnf 11198  df-mnf 11199  df-xr 11200  df-ltxr 11201  df-le 11202  df-sub 11394  df-neg 11395  df-div 11820  df-nn 12161  df-2 12223  df-n0 12421  df-z 12507  df-uz 12771  df-q 12881  df-rp 12923  df-xneg 13040  df-xadd 13041  df-xmul 13042  df-rest 17311  df-topgen 17332  df-psmet 20804  df-xmet 20805  df-bl 20807  df-mopn 20808  df-top 22259  df-topon 22276  df-bases 22312
This theorem is referenced by:  ressxms  23897  nrginvrcn  24072  resubmet  24181  tgioo2  24182  metdscn2  24236  divcn  24247  dfii3  24262  cncfcn  24289  metsscmetcld  24695  cmetss  24696  minveclem4a  24810  ftc1lem6  25421  ulmdvlem3  25777  abelth  25816  cxpcn3  26117  rlimcnp  26331  minvecolem4b  29862  minvecolem4  29864  hhsscms  30262  ftc1cnnc  36179
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