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Theorem caublcls 24206
Description: The convergent point of a sequence of nested balls is in the closures of any of the balls (i.e. it is in the intersection of the closures). Indeed, it is the only point in the intersection because a metric space is Hausdorff, but we don't prove this here. (Contributed by Mario Carneiro, 21-Jan-2014.) (Revised by Mario Carneiro, 1-May-2014.)
Hypotheses
Ref Expression
caubl.2 (𝜑𝐷 ∈ (∞Met‘𝑋))
caubl.3 (𝜑𝐹:ℕ⟶(𝑋 × ℝ+))
caubl.4 (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))
caublcls.6 𝐽 = (MetOpen‘𝐷)
Assertion
Ref Expression
caublcls ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝑃 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝐹𝐴))))
Distinct variable groups:   𝐷,𝑛   𝑛,𝐹   𝑛,𝑋
Allowed substitution hints:   𝜑(𝑛)   𝐴(𝑛)   𝑃(𝑛)   𝐽(𝑛)

Proof of Theorem caublcls
Dummy variables 𝑘 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . 2 (ℤ𝐴) = (ℤ𝐴)
2 caubl.2 . . . 4 (𝜑𝐷 ∈ (∞Met‘𝑋))
323ad2ant1 1135 . . 3 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝐷 ∈ (∞Met‘𝑋))
4 caublcls.6 . . . 4 𝐽 = (MetOpen‘𝐷)
54mopntopon 23337 . . 3 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
63, 5syl 17 . 2 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝐽 ∈ (TopOn‘𝑋))
7 simp3 1140 . . 3 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝐴 ∈ ℕ)
87nnzd 12281 . 2 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝐴 ∈ ℤ)
9 simp2 1139 . 2 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (1st𝐹)(⇝𝑡𝐽)𝑃)
10 2fveq3 6722 . . . . . . . 8 (𝑟 = 𝐴 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝐴)))
1110sseq1d 3932 . . . . . . 7 (𝑟 = 𝐴 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) ↔ ((ball‘𝐷)‘(𝐹𝐴)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
1211imbi2d 344 . . . . . 6 (𝑟 = 𝐴 → (((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))) ↔ ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
13 2fveq3 6722 . . . . . . . 8 (𝑟 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹𝑘)))
1413sseq1d 3932 . . . . . . 7 (𝑟 = 𝑘 → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) ↔ ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
1514imbi2d 344 . . . . . 6 (𝑟 = 𝑘 → (((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))) ↔ ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
16 2fveq3 6722 . . . . . . . 8 (𝑟 = (𝑘 + 1) → ((ball‘𝐷)‘(𝐹𝑟)) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
1716sseq1d 3932 . . . . . . 7 (𝑟 = (𝑘 + 1) → (((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
1817imbi2d 344 . . . . . 6 (𝑟 = (𝑘 + 1) → (((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑟)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))) ↔ ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
19 ssid 3923 . . . . . . 7 ((ball‘𝐷)‘(𝐹𝐴)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))
20192a1i 12 . . . . . 6 (𝐴 ∈ ℤ → ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
21 caubl.4 . . . . . . . . . . 11 (𝜑 → ∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)))
22 eluznn 12514 . . . . . . . . . . 11 ((𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝐴)) → 𝑘 ∈ ℕ)
23 fvoveq1 7236 . . . . . . . . . . . . . 14 (𝑛 = 𝑘 → (𝐹‘(𝑛 + 1)) = (𝐹‘(𝑘 + 1)))
2423fveq2d 6721 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) = ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))))
25 2fveq3 6722 . . . . . . . . . . . . 13 (𝑛 = 𝑘 → ((ball‘𝐷)‘(𝐹𝑛)) = ((ball‘𝐷)‘(𝐹𝑘)))
2624, 25sseq12d 3934 . . . . . . . . . . . 12 (𝑛 = 𝑘 → (((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ↔ ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘))))
2726rspccva 3536 . . . . . . . . . . 11 ((∀𝑛 ∈ ℕ ((ball‘𝐷)‘(𝐹‘(𝑛 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑛)) ∧ 𝑘 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
2821, 22, 27syl2an 599 . . . . . . . . . 10 ((𝜑 ∧ (𝐴 ∈ ℕ ∧ 𝑘 ∈ (ℤ𝐴))) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
2928anassrs 471 . . . . . . . . 9 (((𝜑𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)))
30 sstr2 3908 . . . . . . . . 9 (((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝑘)) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
3129, 30syl 17 . . . . . . . 8 (((𝜑𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
3231expcom 417 . . . . . . 7 (𝑘 ∈ (ℤ𝐴) → ((𝜑𝐴 ∈ ℕ) → (((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
3332a2d 29 . . . . . 6 (𝑘 ∈ (ℤ𝐴) → (((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))) → ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹‘(𝑘 + 1))) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))))
3412, 15, 18, 15, 20, 33uzind4 12502 . . . . 5 (𝑘 ∈ (ℤ𝐴) → ((𝜑𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴))))
3534impcom 411 . . . 4 (((𝜑𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))
36353adantl2 1169 . . 3 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹𝑘)) ⊆ ((ball‘𝐷)‘(𝐹𝐴)))
373adantr 484 . . . . 5 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → 𝐷 ∈ (∞Met‘𝑋))
38 simpl1 1193 . . . . . . . 8 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → 𝜑)
39 caubl.3 . . . . . . . 8 (𝜑𝐹:ℕ⟶(𝑋 × ℝ+))
4038, 39syl 17 . . . . . . 7 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → 𝐹:ℕ⟶(𝑋 × ℝ+))
41223ad2antl3 1189 . . . . . . 7 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → 𝑘 ∈ ℕ)
4240, 41ffvelrnd 6905 . . . . . 6 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (𝐹𝑘) ∈ (𝑋 × ℝ+))
43 xp1st 7793 . . . . . 6 ((𝐹𝑘) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹𝑘)) ∈ 𝑋)
4442, 43syl 17 . . . . 5 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (1st ‘(𝐹𝑘)) ∈ 𝑋)
45 xp2nd 7794 . . . . . 6 ((𝐹𝑘) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹𝑘)) ∈ ℝ+)
4642, 45syl 17 . . . . 5 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (2nd ‘(𝐹𝑘)) ∈ ℝ+)
47 blcntr 23311 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹𝑘)) ∈ 𝑋 ∧ (2nd ‘(𝐹𝑘)) ∈ ℝ+) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
4837, 44, 46, 47syl3anc 1373 . . . 4 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (1st ‘(𝐹𝑘)) ∈ ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
49 fvco3 6810 . . . . 5 ((𝐹:ℕ⟶(𝑋 × ℝ+) ∧ 𝑘 ∈ ℕ) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
5040, 41, 49syl2anc 587 . . . 4 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((1st𝐹)‘𝑘) = (1st ‘(𝐹𝑘)))
51 1st2nd2 7800 . . . . . . 7 ((𝐹𝑘) ∈ (𝑋 × ℝ+) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
5242, 51syl 17 . . . . . 6 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → (𝐹𝑘) = ⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
5352fveq2d 6721 . . . . 5 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹𝑘)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩))
54 df-ov 7216 . . . . 5 ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝑘)), (2nd ‘(𝐹𝑘))⟩)
5553, 54eqtr4di 2796 . . . 4 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((ball‘𝐷)‘(𝐹𝑘)) = ((1st ‘(𝐹𝑘))(ball‘𝐷)(2nd ‘(𝐹𝑘))))
5648, 50, 553eltr4d 2853 . . 3 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((1st𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹𝑘)))
5736, 56sseldd 3902 . 2 (((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) ∧ 𝑘 ∈ (ℤ𝐴)) → ((1st𝐹)‘𝑘) ∈ ((ball‘𝐷)‘(𝐹𝐴)))
5839ffvelrnda 6904 . . . . . . 7 ((𝜑𝐴 ∈ ℕ) → (𝐹𝐴) ∈ (𝑋 × ℝ+))
59583adant2 1133 . . . . . 6 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (𝐹𝐴) ∈ (𝑋 × ℝ+))
60 1st2nd2 7800 . . . . . 6 ((𝐹𝐴) ∈ (𝑋 × ℝ+) → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
6159, 60syl 17 . . . . 5 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (𝐹𝐴) = ⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
6261fveq2d 6721 . . . 4 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩))
63 df-ov 7216 . . . 4 ((1st ‘(𝐹𝐴))(ball‘𝐷)(2nd ‘(𝐹𝐴))) = ((ball‘𝐷)‘⟨(1st ‘(𝐹𝐴)), (2nd ‘(𝐹𝐴))⟩)
6462, 63eqtr4di 2796 . . 3 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) = ((1st ‘(𝐹𝐴))(ball‘𝐷)(2nd ‘(𝐹𝐴))))
65 xp1st 7793 . . . . 5 ((𝐹𝐴) ∈ (𝑋 × ℝ+) → (1st ‘(𝐹𝐴)) ∈ 𝑋)
6659, 65syl 17 . . . 4 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (1st ‘(𝐹𝐴)) ∈ 𝑋)
67 xp2nd 7794 . . . . . 6 ((𝐹𝐴) ∈ (𝑋 × ℝ+) → (2nd ‘(𝐹𝐴)) ∈ ℝ+)
6859, 67syl 17 . . . . 5 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (2nd ‘(𝐹𝐴)) ∈ ℝ+)
6968rpxrd 12629 . . . 4 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → (2nd ‘(𝐹𝐴)) ∈ ℝ*)
70 blssm 23316 . . . 4 ((𝐷 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝐹𝐴)) ∈ 𝑋 ∧ (2nd ‘(𝐹𝐴)) ∈ ℝ*) → ((1st ‘(𝐹𝐴))(ball‘𝐷)(2nd ‘(𝐹𝐴))) ⊆ 𝑋)
713, 66, 69, 70syl3anc 1373 . . 3 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → ((1st ‘(𝐹𝐴))(ball‘𝐷)(2nd ‘(𝐹𝐴))) ⊆ 𝑋)
7264, 71eqsstrd 3939 . 2 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → ((ball‘𝐷)‘(𝐹𝐴)) ⊆ 𝑋)
731, 6, 8, 9, 57, 72lmcls 22199 1 ((𝜑 ∧ (1st𝐹)(⇝𝑡𝐽)𝑃𝐴 ∈ ℕ) → 𝑃 ∈ ((cls‘𝐽)‘((ball‘𝐷)‘(𝐹𝐴))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  wss 3866  cop 4547   class class class wbr 5053   × cxp 5549  ccom 5555  wf 6376  cfv 6380  (class class class)co 7213  1st c1st 7759  2nd c2nd 7760  1c1 10730   + caddc 10732  *cxr 10866  cn 11830  cz 12176  cuz 12438  +crp 12586  ∞Metcxmet 20348  ballcbl 20350  MetOpencmopn 20353  TopOnctopon 21807  clsccl 21915  𝑡clm 22123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806  ax-pre-sup 10807
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-er 8391  df-map 8510  df-pm 8511  df-en 8627  df-dom 8628  df-sdom 8629  df-sup 9058  df-inf 9059  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-n0 12091  df-z 12177  df-uz 12439  df-q 12545  df-rp 12587  df-xneg 12704  df-xadd 12705  df-xmul 12706  df-topgen 16948  df-psmet 20355  df-xmet 20356  df-bl 20358  df-mopn 20359  df-top 21791  df-topon 21808  df-bases 21843  df-cld 21916  df-ntr 21917  df-cls 21918  df-lm 22126
This theorem is referenced by:  bcthlem3  24223  heiborlem8  35713
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