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Theorem bcth 24837
Description: Baire's Category Theorem. If a nonempty metric space is complete, it is nonmeager in itself. In other words, no open set in the metric space can be the countable union of rare closed subsets (where rare means having a closure with empty interior), so some subset π‘€β€˜π‘˜ must have a nonempty interior. Theorem 4.7-2 of [Kreyszig] p. 247. (The terminology "meager" and "nonmeager" is used by Kreyszig to replace Baire's "of the first category" and "of the second category." The latter terms are going out of favor to avoid confusion with category theory.) See bcthlem5 24836 for an overview of the proof. (Contributed by NM, 28-Oct-2007.) (Proof shortened by Mario Carneiro, 6-Jan-2014.)
Hypothesis
Ref Expression
bcth.2 𝐽 = (MetOpenβ€˜π·)
Assertion
Ref Expression
bcth ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑀:β„•βŸΆ(Clsdβ€˜π½) ∧ ((intβ€˜π½)β€˜βˆͺ ran 𝑀) β‰  βˆ…) β†’ βˆƒπ‘˜ ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) β‰  βˆ…)
Distinct variable groups:   𝐷,π‘˜   π‘˜,𝐽   π‘˜,𝑀   π‘˜,𝑋
Proof of Theorem bcth
Dummy variables 𝑛 π‘Ÿ π‘₯ 𝑧 𝑔 π‘š 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bcth.2 . . . . . 6 𝐽 = (MetOpenβ€˜π·)
2 simpll 765 . . . . . 6 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑀:β„•βŸΆ(Clsdβ€˜π½)) ∧ βˆ€π‘˜ ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ…) β†’ 𝐷 ∈ (CMetβ€˜π‘‹))
3 eleq1w 2816 . . . . . . . . . . 11 (π‘₯ = 𝑦 β†’ (π‘₯ ∈ 𝑋 ↔ 𝑦 ∈ 𝑋))
4 eleq1w 2816 . . . . . . . . . . 11 (π‘Ÿ = π‘š β†’ (π‘Ÿ ∈ ℝ+ ↔ π‘š ∈ ℝ+))
53, 4bi2anan9 637 . . . . . . . . . 10 ((π‘₯ = 𝑦 ∧ π‘Ÿ = π‘š) β†’ ((π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) ↔ (𝑦 ∈ 𝑋 ∧ π‘š ∈ ℝ+)))
6 simpr 485 . . . . . . . . . . . 12 ((π‘₯ = 𝑦 ∧ π‘Ÿ = π‘š) β†’ π‘Ÿ = π‘š)
76breq1d 5157 . . . . . . . . . . 11 ((π‘₯ = 𝑦 ∧ π‘Ÿ = π‘š) β†’ (π‘Ÿ < (1 / π‘˜) ↔ π‘š < (1 / π‘˜)))
8 oveq12 7414 . . . . . . . . . . . . 13 ((π‘₯ = 𝑦 ∧ π‘Ÿ = π‘š) β†’ (π‘₯(ballβ€˜π·)π‘Ÿ) = (𝑦(ballβ€˜π·)π‘š))
98fveq2d 6892 . . . . . . . . . . . 12 ((π‘₯ = 𝑦 ∧ π‘Ÿ = π‘š) β†’ ((clsβ€˜π½)β€˜(π‘₯(ballβ€˜π·)π‘Ÿ)) = ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)))
109sseq1d 4012 . . . . . . . . . . 11 ((π‘₯ = 𝑦 ∧ π‘Ÿ = π‘š) β†’ (((clsβ€˜π½)β€˜(π‘₯(ballβ€˜π·)π‘Ÿ)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜)) ↔ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))))
117, 10anbi12d 631 . . . . . . . . . 10 ((π‘₯ = 𝑦 ∧ π‘Ÿ = π‘š) β†’ ((π‘Ÿ < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(π‘₯(ballβ€˜π·)π‘Ÿ)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))) ↔ (π‘š < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜)))))
125, 11anbi12d 631 . . . . . . . . 9 ((π‘₯ = 𝑦 ∧ π‘Ÿ = π‘š) β†’ (((π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) ∧ (π‘Ÿ < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(π‘₯(ballβ€˜π·)π‘Ÿ)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜)))) ↔ ((𝑦 ∈ 𝑋 ∧ π‘š ∈ ℝ+) ∧ (π‘š < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))))))
1312cbvopabv 5220 . . . . . . . 8 {⟨π‘₯, π‘ŸβŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) ∧ (π‘Ÿ < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(π‘₯(ballβ€˜π·)π‘Ÿ)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))))} = {βŸ¨π‘¦, π‘šβŸ© ∣ ((𝑦 ∈ 𝑋 ∧ π‘š ∈ ℝ+) ∧ (π‘š < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))))}
14 oveq2 7413 . . . . . . . . . . . 12 (π‘˜ = 𝑛 β†’ (1 / π‘˜) = (1 / 𝑛))
1514breq2d 5159 . . . . . . . . . . 11 (π‘˜ = 𝑛 β†’ (π‘š < (1 / π‘˜) ↔ π‘š < (1 / 𝑛)))
16 fveq2 6888 . . . . . . . . . . . . 13 (π‘˜ = 𝑛 β†’ (π‘€β€˜π‘˜) = (π‘€β€˜π‘›))
1716difeq2d 4121 . . . . . . . . . . . 12 (π‘˜ = 𝑛 β†’ (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜)) = (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘›)))
1817sseq2d 4013 . . . . . . . . . . 11 (π‘˜ = 𝑛 β†’ (((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜)) ↔ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘›))))
1915, 18anbi12d 631 . . . . . . . . . 10 (π‘˜ = 𝑛 β†’ ((π‘š < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))) ↔ (π‘š < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘›)))))
2019anbi2d 629 . . . . . . . . 9 (π‘˜ = 𝑛 β†’ (((𝑦 ∈ 𝑋 ∧ π‘š ∈ ℝ+) ∧ (π‘š < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜)))) ↔ ((𝑦 ∈ 𝑋 ∧ π‘š ∈ ℝ+) ∧ (π‘š < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘›))))))
2120opabbidv 5213 . . . . . . . 8 (π‘˜ = 𝑛 β†’ {βŸ¨π‘¦, π‘šβŸ© ∣ ((𝑦 ∈ 𝑋 ∧ π‘š ∈ ℝ+) ∧ (π‘š < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))))} = {βŸ¨π‘¦, π‘šβŸ© ∣ ((𝑦 ∈ 𝑋 ∧ π‘š ∈ ℝ+) ∧ (π‘š < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘›))))})
2213, 21eqtrid 2784 . . . . . . 7 (π‘˜ = 𝑛 β†’ {⟨π‘₯, π‘ŸβŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) ∧ (π‘Ÿ < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(π‘₯(ballβ€˜π·)π‘Ÿ)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))))} = {βŸ¨π‘¦, π‘šβŸ© ∣ ((𝑦 ∈ 𝑋 ∧ π‘š ∈ ℝ+) ∧ (π‘š < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘›))))})
23 fveq2 6888 . . . . . . . . . . . 12 (𝑧 = 𝑔 β†’ ((ballβ€˜π·)β€˜π‘§) = ((ballβ€˜π·)β€˜π‘”))
2423difeq1d 4120 . . . . . . . . . . 11 (𝑧 = 𝑔 β†’ (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘›)) = (((ballβ€˜π·)β€˜π‘”) βˆ– (π‘€β€˜π‘›)))
2524sseq2d 4013 . . . . . . . . . 10 (𝑧 = 𝑔 β†’ (((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘›)) ↔ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘”) βˆ– (π‘€β€˜π‘›))))
2625anbi2d 629 . . . . . . . . 9 (𝑧 = 𝑔 β†’ ((π‘š < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘›))) ↔ (π‘š < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘”) βˆ– (π‘€β€˜π‘›)))))
2726anbi2d 629 . . . . . . . 8 (𝑧 = 𝑔 β†’ (((𝑦 ∈ 𝑋 ∧ π‘š ∈ ℝ+) ∧ (π‘š < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘›)))) ↔ ((𝑦 ∈ 𝑋 ∧ π‘š ∈ ℝ+) ∧ (π‘š < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘”) βˆ– (π‘€β€˜π‘›))))))
2827opabbidv 5213 . . . . . . 7 (𝑧 = 𝑔 β†’ {βŸ¨π‘¦, π‘šβŸ© ∣ ((𝑦 ∈ 𝑋 ∧ π‘š ∈ ℝ+) ∧ (π‘š < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘›))))} = {βŸ¨π‘¦, π‘šβŸ© ∣ ((𝑦 ∈ 𝑋 ∧ π‘š ∈ ℝ+) ∧ (π‘š < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘”) βˆ– (π‘€β€˜π‘›))))})
2922, 28cbvmpov 7500 . . . . . 6 (π‘˜ ∈ β„•, 𝑧 ∈ (𝑋 Γ— ℝ+) ↦ {⟨π‘₯, π‘ŸβŸ© ∣ ((π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) ∧ (π‘Ÿ < (1 / π‘˜) ∧ ((clsβ€˜π½)β€˜(π‘₯(ballβ€˜π·)π‘Ÿ)) βŠ† (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜))))}) = (𝑛 ∈ β„•, 𝑔 ∈ (𝑋 Γ— ℝ+) ↦ {βŸ¨π‘¦, π‘šβŸ© ∣ ((𝑦 ∈ 𝑋 ∧ π‘š ∈ ℝ+) ∧ (π‘š < (1 / 𝑛) ∧ ((clsβ€˜π½)β€˜(𝑦(ballβ€˜π·)π‘š)) βŠ† (((ballβ€˜π·)β€˜π‘”) βˆ– (π‘€β€˜π‘›))))})
30 simplr 767 . . . . . 6 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑀:β„•βŸΆ(Clsdβ€˜π½)) ∧ βˆ€π‘˜ ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ…) β†’ 𝑀:β„•βŸΆ(Clsdβ€˜π½))
31 simpr 485 . . . . . . 7 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑀:β„•βŸΆ(Clsdβ€˜π½)) ∧ βˆ€π‘˜ ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ…) β†’ βˆ€π‘˜ ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ…)
3216fveqeq2d 6896 . . . . . . . 8 (π‘˜ = 𝑛 β†’ (((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ… ↔ ((intβ€˜π½)β€˜(π‘€β€˜π‘›)) = βˆ…))
3332cbvralvw 3234 . . . . . . 7 (βˆ€π‘˜ ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ… ↔ βˆ€π‘› ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘›)) = βˆ…)
3431, 33sylib 217 . . . . . 6 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑀:β„•βŸΆ(Clsdβ€˜π½)) ∧ βˆ€π‘˜ ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ…) β†’ βˆ€π‘› ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘›)) = βˆ…)
351, 2, 29, 30, 34bcthlem5 24836 . . . . 5 (((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑀:β„•βŸΆ(Clsdβ€˜π½)) ∧ βˆ€π‘˜ ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ…) β†’ ((intβ€˜π½)β€˜βˆͺ ran 𝑀) = βˆ…)
3635ex 413 . . . 4 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑀:β„•βŸΆ(Clsdβ€˜π½)) β†’ (βˆ€π‘˜ ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ… β†’ ((intβ€˜π½)β€˜βˆͺ ran 𝑀) = βˆ…))
3736necon3ad 2953 . . 3 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑀:β„•βŸΆ(Clsdβ€˜π½)) β†’ (((intβ€˜π½)β€˜βˆͺ ran 𝑀) β‰  βˆ… β†’ Β¬ βˆ€π‘˜ ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ…))
38373impia 1117 . 2 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑀:β„•βŸΆ(Clsdβ€˜π½) ∧ ((intβ€˜π½)β€˜βˆͺ ran 𝑀) β‰  βˆ…) β†’ Β¬ βˆ€π‘˜ ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ…)
39 df-ne 2941 . . . 4 (((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) β‰  βˆ… ↔ Β¬ ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ…)
4039rexbii 3094 . . 3 (βˆƒπ‘˜ ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) β‰  βˆ… ↔ βˆƒπ‘˜ ∈ β„• Β¬ ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ…)
41 rexnal 3100 . . 3 (βˆƒπ‘˜ ∈ β„• Β¬ ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ… ↔ Β¬ βˆ€π‘˜ ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ…)
4240, 41bitri 274 . 2 (βˆƒπ‘˜ ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) β‰  βˆ… ↔ Β¬ βˆ€π‘˜ ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ…)
4338, 42sylibr 233 1 ((𝐷 ∈ (CMetβ€˜π‘‹) ∧ 𝑀:β„•βŸΆ(Clsdβ€˜π½) ∧ ((intβ€˜π½)β€˜βˆͺ ran 𝑀) β‰  βˆ…) β†’ βˆƒπ‘˜ ∈ β„• ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) β‰  βˆ…)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070   βˆ– cdif 3944   βŠ† wss 3947  βˆ…c0 4321  βˆͺ cuni 4907   class class class wbr 5147  {copab 5209   Γ— cxp 5673  ran crn 5676  βŸΆwf 6536  β€˜cfv 6540  (class class class)co 7405   ∈ cmpo 7407  1c1 11107   < clt 11244   / cdiv 11867  β„•cn 12208  β„+crp 12970  ballcbl 20923  MetOpencmopn 20926  Clsdccld 22511  intcnt 22512  clsccl 22513  CMetccmet 24762
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 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721  ax-inf2 9632  ax-dc 10437  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
This theorem depends on definitions:  df-bi 206  df-an 397  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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7852  df-1st 7971  df-2nd 7972  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-1o 8462  df-er 8699  df-map 8818  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-sup 9433  df-inf 9434  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-n0 12469  df-z 12555  df-uz 12819  df-q 12929  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-ico 13326  df-rest 17364  df-topgen 17385  df-psmet 20928  df-xmet 20929  df-met 20930  df-bl 20931  df-mopn 20932  df-fbas 20933  df-fg 20934  df-top 22387  df-topon 22404  df-bases 22440  df-cld 22514  df-ntr 22515  df-cls 22516  df-nei 22593  df-lm 22724  df-fil 23341  df-fm 23433  df-flim 23434  df-flf 23435  df-cfil 24763  df-cau 24764  df-cmet 24765
This theorem is referenced by:  bcth2  24838  bcth3  24839
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