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Theorem metrest 14045
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 3357 . . . . . . . . . 10 (𝑒 ∩ π‘Œ) βŠ† 𝑒
2 metrest.3 . . . . . . . . . . . . 13 𝐽 = (MetOpenβ€˜πΆ)
32elmopn2 13988 . . . . . . . . . . . 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 384 . . . . . . . . . . 11 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑒 ∈ 𝐽) β†’ βˆ€π‘¦ ∈ 𝑒 βˆƒπ‘Ÿ ∈ ℝ+ (𝑦(ballβ€˜πΆ)π‘Ÿ) βŠ† 𝑒)
54adantlr 477 . . . . . . . . . 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 3221 . . . . . . . . . 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 65 . . . . . . . . 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 3362 . . . . . . . . . . 11 ((𝑦(ballβ€˜πΆ)π‘Ÿ) βŠ† 𝑒 β†’ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† (𝑒 ∩ π‘Œ))
98reximi 2574 . . . . . . . . . 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 2540 . . . . . . . . 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 14 . . . . . . . 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 3358 . . . . . . . 8 (𝑒 ∩ π‘Œ) βŠ† π‘Œ
1311, 12jctil 312 . . . . . . 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 3180 . . . . . . . 8 (π‘₯ = (𝑒 ∩ π‘Œ) β†’ (π‘₯ βŠ† π‘Œ ↔ (𝑒 ∩ π‘Œ) βŠ† π‘Œ))
15 sseq2 3181 . . . . . . . . . 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 2478 . . . . . . . . 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 2681 . . . . . . . 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 473 . . . . . . 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 157 . . . . . 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 2594 . . . . 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 13983 . . . . . . . . 9 (𝐢 ∈ (∞Metβ€˜π‘‹) β†’ 𝐽 ∈ Top)
2221ad2antrr 488 . . . . . . . 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 3152 . . . . . . . . . . . . . 14 ((π‘₯ βŠ† π‘Œ ∧ 𝑦 ∈ π‘₯) β†’ 𝑦 ∈ π‘Œ)
24 ssel2 3152 . . . . . . . . . . . . . . . 16 ((π‘Œ βŠ† 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝑦 ∈ 𝑋)
25 rpxr 9663 . . . . . . . . . . . . . . . . . 18 (π‘Ÿ ∈ ℝ+ β†’ π‘Ÿ ∈ ℝ*)
262blopn 14029 . . . . . . . . . . . . . . . . . . . 20 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑦(ballβ€˜πΆ)π‘Ÿ) ∈ 𝐽)
27 eleq1a 2249 . . . . . . . . . . . . . . . . . . . 20 ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∈ 𝐽 β†’ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
2826, 27syl 14 . . . . . . . . . . . . . . . . . . 19 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
29283expa 1203 . . . . . . . . . . . . . . . . . 18 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ π‘Ÿ ∈ ℝ*) β†’ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
3025, 29sylan2 286 . . . . . . . . . . . . . . . . 17 (((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) ∧ π‘Ÿ ∈ ℝ+) β†’ (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
3130rexlimdva 2594 . . . . . . . . . . . . . . . 16 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑦 ∈ 𝑋) β†’ (βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ 𝑧 ∈ 𝐽))
3224, 31sylan2 286 . . . . . . . . . . . . . . 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 400 . . . . . . . . . . . . . 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 286 . . . . . . . . . . . . 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 400 . . . . . . . . . . . 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 2594 . . . . . . . . . . 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 279 . . . . . . . . . 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 479 . . . . . . . . 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 3231 . . . . . . . 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 13540 . . . . . . . 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 411 . . . . . . 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 5884 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑒 β†’ (𝑦(ballβ€˜πΆ)π‘Ÿ) = (𝑒(ballβ€˜πΆ)π‘Ÿ))
4342ineq1d 3337 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑒 β†’ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) = ((𝑒(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ))
4443sseq1d 3186 . . . . . . . . . . . . . . . 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 2478 . . . . . . . . . . . . . . 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 2840 . . . . . . . . . . . . . 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 491 . . . . . . . . . . . . 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 3151 . . . . . . . . . . . . . . 15 (π‘₯ βŠ† π‘Œ β†’ (𝑒 ∈ π‘₯ β†’ 𝑒 ∈ π‘Œ))
49 ssel 3151 . . . . . . . . . . . . . . . 16 (π‘Œ βŠ† 𝑋 β†’ (𝑒 ∈ π‘Œ β†’ 𝑒 ∈ 𝑋))
50 blcntr 13955 . . . . . . . . . . . . . . . . . . . . 21 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ 𝑒 ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) β†’ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ))
5150a1d 22 . . . . . . . . . . . . . . . . . . . 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 325 . . . . . . . . . . . . . . . . . . 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 1203 . . . . . . . . . . . . . . . . . 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 2579 . . . . . . . . . . . . . . . . 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 115 . . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . . 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 410 . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 41 . . . . . . . . . . . 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 2247 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑒 β†’ (𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ) ↔ 𝑒 ∈ (𝑒(ballβ€˜πΆ)π‘Ÿ)))
6144, 60anbi12d 473 . . . . . . . . . . . . . . 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 2478 . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . 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 115 . . . . . . . . . . . 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 28 . . . . . . . . . . 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 529 . . . . . . . . . . . 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 3156 . . . . . . . . . . 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 307 . . . . . . . . . 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 3320 . . . . . . . . . . . . . . 15 (𝑒 ∈ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) ↔ (𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ 𝑒 ∈ π‘Œ))
70 ssel2 3152 . . . . . . . . . . . . . . 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 288 . . . . . . . . . . . . . 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 375 . . . . . . . . . . . . 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 2590 . . . . . . . . . . . 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 2590 . . . . . . . . . . 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 124 . . . . . . . . . 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 142 . . . . . . . . 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 3320 . . . . . . . . . . 11 (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 3823 . . . . . . . . . . . . . 14 (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 266 . . . . . . . . . . . . . . . 16 ((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 401 . . . . . . . . . . . . . . . 16 (((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 2633 . . . . . . . . . . . . . . . . . 18 (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 2484 . . . . . . . . . . . . . . . . 17 (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 2633 . . . . . . . . . . . . . . . . 17 (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 185 . . . . . . . . . . . . . . . 16 ((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 206 . . . . . . . . . . . . . . 15 ((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 1605 . . . . . . . . . . . . . 14 (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)))
8778, 86bitri 184 . . . . . . . . . . . . 13 (𝑒 ∈ βˆͺ {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)))
88 vex 2742 . . . . . . . . . . . . . . . . . . 19 𝑦 ∈ V
89 blex 13926 . . . . . . . . . . . . . . . . . . 19 (𝐢 ∈ (∞Metβ€˜π‘‹) β†’ (ballβ€˜πΆ) ∈ V)
90 vex 2742 . . . . . . . . . . . . . . . . . . . 20 π‘Ÿ ∈ V
9190a1i 9 . . . . . . . . . . . . . . . . . . 19 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ π‘Ÿ ∈ V)
92 ovexg 5911 . . . . . . . . . . . . . . . . . . 19 ((𝑦 ∈ V ∧ (ballβ€˜πΆ) ∈ V ∧ π‘Ÿ ∈ V) β†’ (𝑦(ballβ€˜πΆ)π‘Ÿ) ∈ V)
9388, 89, 91, 92mp3an2ani 1344 . . . . . . . . . . . . . . . . . 18 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝑦(ballβ€˜πΆ)π‘Ÿ) ∈ V)
94 ineq1 3331 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ (𝑧 ∩ π‘Œ) = ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ))
9594sseq1d 3186 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ ((𝑧 ∩ π‘Œ) βŠ† π‘₯ ↔ ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∩ π‘Œ) βŠ† π‘₯))
96 eleq2 2241 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) β†’ (𝑒 ∈ 𝑧 ↔ 𝑒 ∈ (𝑦(ballβ€˜πΆ)π‘Ÿ)))
9795, 96anbi12d 473 . . . . . . . . . . . . . . . . . . 19 (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))))
9897ceqsexgv 2868 . . . . . . . . . . . . . . . . . 18 ((𝑦(ballβ€˜πΆ)π‘Ÿ) ∈ V β†’ (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))))
9993, 98syl 14 . . . . . . . . . . . . . . . . 17 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (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))))
10099rexbidv 2478 . . . . . . . . . . . . . . . 16 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (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))))
101 rexcom4 2762 . . . . . . . . . . . . . . . 16 (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)))
102100, 101bitr3di 195 . . . . . . . . . . . . . . 15 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (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))))
103102rexbidv 2478 . . . . . . . . . . . . . 14 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (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))))
104 rexcom4 2762 . . . . . . . . . . . . . 14 (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)))
105103, 104bitr2di 197 . . . . . . . . . . . . 13 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (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))))
10687, 105bitrid 192 . . . . . . . . . . . 12 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (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))))
107106anbi1d 465 . . . . . . . . . . 11 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ ((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)))
10877, 107bitr2id 193 . . . . . . . . . 10 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) 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) ↔ 𝑒 ∈ (βˆͺ {𝑧 ∣ (βˆƒπ‘¦ ∈ π‘₯ βˆƒπ‘Ÿ ∈ ℝ+ 𝑧 = (𝑦(ballβ€˜πΆ)π‘Ÿ) ∧ (𝑧 ∩ π‘Œ) βŠ† π‘₯)} ∩ π‘Œ)))
109108adantr 276 . . . . . . . . 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+ (((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)))
11076, 109bitrd 188 . . . . . . . 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)))
111110eqrdv 2175 . . . . . . 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))
112 ineq1 3331 . . . . . . . 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))
113112rspceeqv 2861 . . . . . . 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))
11441, 111, 113syl2anc 411 . . . . . 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))
115114ex 115 . . . . 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)))
11620, 115impbid 129 . . . 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‘₯)))
117 simpr 110 . . . . . . . . . . 11 ((π‘Œ βŠ† 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝑦 ∈ π‘Œ)
11824, 117elind 3322 . . . . . . . . . 10 ((π‘Œ βŠ† 𝑋 ∧ 𝑦 ∈ π‘Œ) β†’ 𝑦 ∈ (𝑋 ∩ π‘Œ))
119 metrest.1 . . . . . . . . . . . . . . 15 𝐷 = (𝐢 β†Ύ (π‘Œ Γ— π‘Œ))
120119blres 13973 . . . . . . . . . . . . . 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))
121120sseq1d 3186 . . . . . . . . . . . . 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‘₯))
1221213expa 1203 . . . . . . . . . . . 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‘₯))
12325, 122sylan2 286 . . . . . . . . . . 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‘₯))
124123rexbidva 2474 . . . . . . . . . 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‘₯))
125118, 124sylan2 286 . . . . . . . . 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‘₯))
126125anassrs 400 . . . . . . . 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‘₯))
12723, 126sylan2 286 . . . . . . 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‘₯))
128127anassrs 400 . . . . . 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‘₯))
129128ralbidva 2473 . . . . 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‘₯))
130129pm5.32da 452 . . . 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‘₯)))
131116, 130bitr4d 191 . . 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‘₯)))
13221adantr 276 . . . 4 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ 𝐽 ∈ Top)
133 id 19 . . . . 5 (π‘Œ βŠ† 𝑋 β†’ π‘Œ βŠ† 𝑋)
1342mopnm 13987 . . . . 5 (𝐢 ∈ (∞Metβ€˜π‘‹) β†’ 𝑋 ∈ 𝐽)
135 ssexg 4144 . . . . 5 ((π‘Œ βŠ† 𝑋 ∧ 𝑋 ∈ 𝐽) β†’ π‘Œ ∈ V)
136133, 134, 135syl2anr 290 . . . 4 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ π‘Œ ∈ V)
137 elrest 12700 . . . 4 ((𝐽 ∈ Top ∧ π‘Œ ∈ V) β†’ (π‘₯ ∈ (𝐽 β†Ύt π‘Œ) ↔ βˆƒπ‘’ ∈ 𝐽 π‘₯ = (𝑒 ∩ π‘Œ)))
138132, 136, 137syl2anc 411 . . 3 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (π‘₯ ∈ (𝐽 β†Ύt π‘Œ) ↔ βˆƒπ‘’ ∈ 𝐽 π‘₯ = (𝑒 ∩ π‘Œ)))
139 xmetres2 13918 . . . . 5 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝐢 β†Ύ (π‘Œ Γ— π‘Œ)) ∈ (∞Metβ€˜π‘Œ))
140119, 139eqeltrid 2264 . . . 4 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ 𝐷 ∈ (∞Metβ€˜π‘Œ))
141 metrest.4 . . . . 5 𝐾 = (MetOpenβ€˜π·)
142141elmopn2 13988 . . . 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‘₯)))
143140, 142syl 14 . . 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‘₯)))
144131, 138, 1433bitr4d 220 . 2 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (π‘₯ ∈ (𝐽 β†Ύt π‘Œ) ↔ π‘₯ ∈ 𝐾))
145144eqrdv 2175 1 ((𝐢 ∈ (∞Metβ€˜π‘‹) ∧ π‘Œ βŠ† 𝑋) β†’ (𝐽 β†Ύt π‘Œ) = 𝐾)
Colors of variables: wff set class
Syntax hints:   β†’ wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 978   = wceq 1353  βˆƒwex 1492   ∈ wcel 2148  {cab 2163  βˆ€wral 2455  βˆƒwrex 2456  Vcvv 2739   ∩ cin 3130   βŠ† wss 3131  βˆͺ cuni 3811   Γ— cxp 4626   β†Ύ cres 4630  β€˜cfv 5218  (class class class)co 5877  β„*cxr 7993  β„+crp 9655   β†Ύt crest 12693  βˆžMetcxmet 13479  ballcbl 13481  MetOpencmopn 13484  Topctop 13536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933
This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-frec 6394  df-map 6652  df-sup 6985  df-inf 6986  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-xneg 9774  df-xadd 9775  df-seqfrec 10448  df-exp 10522  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-rest 12695  df-topgen 12714  df-psmet 13486  df-xmet 13487  df-bl 13489  df-mopn 13490  df-top 13537  df-topon 13550  df-bases 13582
This theorem is referenced by:  resubmet  14087  tgioo2cntop  14088  divcnap  14094  cncfcncntop  14119  limcimolemlt  14172  cnplimcim  14175  cnplimclemr  14177  limccnpcntop  14183  limccnp2cntop  14185
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