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Theorem bcth 24846
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 24845 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 766 . . . . . 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 2817 . . . . . . . . . . 11 (π‘₯ = 𝑦 β†’ (π‘₯ ∈ 𝑋 ↔ 𝑦 ∈ 𝑋))
4 eleq1w 2817 . . . . . . . . . . 11 (π‘Ÿ = π‘š β†’ (π‘Ÿ ∈ ℝ+ ↔ π‘š ∈ ℝ+))
53, 4bi2anan9 638 . . . . . . . . . 10 ((π‘₯ = 𝑦 ∧ π‘Ÿ = π‘š) β†’ ((π‘₯ ∈ 𝑋 ∧ π‘Ÿ ∈ ℝ+) ↔ (𝑦 ∈ 𝑋 ∧ π‘š ∈ ℝ+)))
6 simpr 486 . . . . . . . . . . . 12 ((π‘₯ = 𝑦 ∧ π‘Ÿ = π‘š) β†’ π‘Ÿ = π‘š)
76breq1d 5159 . . . . . . . . . . 11 ((π‘₯ = 𝑦 ∧ π‘Ÿ = π‘š) β†’ (π‘Ÿ < (1 / π‘˜) ↔ π‘š < (1 / π‘˜)))
8 oveq12 7418 . . . . . . . . . . . . 13 ((π‘₯ = 𝑦 ∧ π‘Ÿ = π‘š) β†’ (π‘₯(ballβ€˜π·)π‘Ÿ) = (𝑦(ballβ€˜π·)π‘š))
98fveq2d 6896 . . . . . . . . . . . 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 4014 . . . . . . . . . . 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 632 . . . . . . . . . 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 632 . . . . . . . . 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 5222 . . . . . . . 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 7417 . . . . . . . . . . . 12 (π‘˜ = 𝑛 β†’ (1 / π‘˜) = (1 / 𝑛))
1514breq2d 5161 . . . . . . . . . . 11 (π‘˜ = 𝑛 β†’ (π‘š < (1 / π‘˜) ↔ π‘š < (1 / 𝑛)))
16 fveq2 6892 . . . . . . . . . . . . 13 (π‘˜ = 𝑛 β†’ (π‘€β€˜π‘˜) = (π‘€β€˜π‘›))
1716difeq2d 4123 . . . . . . . . . . . 12 (π‘˜ = 𝑛 β†’ (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘˜)) = (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘›)))
1817sseq2d 4015 . . . . . . . . . . 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 632 . . . . . . . . . 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 630 . . . . . . . . 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 5215 . . . . . . . 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 2785 . . . . . . 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 6892 . . . . . . . . . . . 12 (𝑧 = 𝑔 β†’ ((ballβ€˜π·)β€˜π‘§) = ((ballβ€˜π·)β€˜π‘”))
2423difeq1d 4122 . . . . . . . . . . 11 (𝑧 = 𝑔 β†’ (((ballβ€˜π·)β€˜π‘§) βˆ– (π‘€β€˜π‘›)) = (((ballβ€˜π·)β€˜π‘”) βˆ– (π‘€β€˜π‘›)))
2524sseq2d 4015 . . . . . . . . . 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 630 . . . . . . . . 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 630 . . . . . . . 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 5215 . . . . . . 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 7504 . . . . . 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 768 . . . . . 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 486 . . . . . . 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 6900 . . . . . . . 8 (π‘˜ = 𝑛 β†’ (((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ… ↔ ((intβ€˜π½)β€˜(π‘€β€˜π‘›)) = βˆ…))
3332cbvralvw 3235 . . . . . . 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 24845 . . . . 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 414 . . . 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 2954 . . 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 1118 . 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 2942 . . . 4 (((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) β‰  βˆ… ↔ Β¬ ((intβ€˜π½)β€˜(π‘€β€˜π‘˜)) = βˆ…)
4039rexbii 3095 . . 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 3101 . . 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 275 . 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 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071   βˆ– cdif 3946   βŠ† wss 3949  βˆ…c0 4323  βˆͺ cuni 4909   class class class wbr 5149  {copab 5211   Γ— cxp 5675  ran crn 5678  βŸΆwf 6540  β€˜cfv 6544  (class class class)co 7409   ∈ cmpo 7411  1c1 11111   < clt 11248   / cdiv 11871  β„•cn 12212  β„+crp 12974  ballcbl 20931  MetOpencmopn 20934  Clsdccld 22520  intcnt 22521  clsccl 22522  CMetccmet 24771
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-inf2 9636  ax-dc 10441  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187  ax-pre-sup 11188
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-er 8703  df-map 8822  df-pm 8823  df-en 8940  df-dom 8941  df-sdom 8942  df-sup 9437  df-inf 9438  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-div 11872  df-nn 12213  df-2 12275  df-n0 12473  df-z 12559  df-uz 12823  df-q 12933  df-rp 12975  df-xneg 13092  df-xadd 13093  df-xmul 13094  df-ico 13330  df-rest 17368  df-topgen 17389  df-psmet 20936  df-xmet 20937  df-met 20938  df-bl 20939  df-mopn 20940  df-fbas 20941  df-fg 20942  df-top 22396  df-topon 22413  df-bases 22449  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602  df-lm 22733  df-fil 23350  df-fm 23442  df-flim 23443  df-flf 23444  df-cfil 24772  df-cau 24773  df-cmet 24774
This theorem is referenced by:  bcth2  24847  bcth3  24848
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