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Theorem caussi 22818
Description: Cauchy sequence on a metric subspace. (Contributed by NM, 30-Jan-2008.) (Revised by Mario Carneiro, 30-Dec-2013.)
Assertion
Ref Expression
caussi (𝐷 ∈ (∞Met‘𝑋) → (Cau‘(𝐷 ↾ (𝑌 × 𝑌))) ⊆ (Cau‘𝐷))

Proof of Theorem caussi
Dummy variables 𝑥 𝑓 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inss1 3791 . . . . . . . . 9 (𝑋𝑌) ⊆ 𝑋
2 xpss2 5138 . . . . . . . . 9 ((𝑋𝑌) ⊆ 𝑋 → (ℂ × (𝑋𝑌)) ⊆ (ℂ × 𝑋))
31, 2ax-mp 5 . . . . . . . 8 (ℂ × (𝑋𝑌)) ⊆ (ℂ × 𝑋)
4 sstr 3572 . . . . . . . 8 ((𝑓 ⊆ (ℂ × (𝑋𝑌)) ∧ (ℂ × (𝑋𝑌)) ⊆ (ℂ × 𝑋)) → 𝑓 ⊆ (ℂ × 𝑋))
53, 4mpan2 702 . . . . . . 7 (𝑓 ⊆ (ℂ × (𝑋𝑌)) → 𝑓 ⊆ (ℂ × 𝑋))
65anim2i 590 . . . . . 6 ((Fun 𝑓𝑓 ⊆ (ℂ × (𝑋𝑌))) → (Fun 𝑓𝑓 ⊆ (ℂ × 𝑋)))
76a1i 11 . . . . 5 (𝐷 ∈ (∞Met‘𝑋) → ((Fun 𝑓𝑓 ⊆ (ℂ × (𝑋𝑌))) → (Fun 𝑓𝑓 ⊆ (ℂ × 𝑋))))
8 elfvdm 6112 . . . . . . 7 (𝐷 ∈ (∞Met‘𝑋) → 𝑋 ∈ dom ∞Met)
9 inex1g 4721 . . . . . . 7 (𝑋 ∈ dom ∞Met → (𝑋𝑌) ∈ V)
108, 9syl 17 . . . . . 6 (𝐷 ∈ (∞Met‘𝑋) → (𝑋𝑌) ∈ V)
11 cnex 9870 . . . . . 6 ℂ ∈ V
12 elpmg 7733 . . . . . 6 (((𝑋𝑌) ∈ V ∧ ℂ ∈ V) → (𝑓 ∈ ((𝑋𝑌) ↑pm ℂ) ↔ (Fun 𝑓𝑓 ⊆ (ℂ × (𝑋𝑌)))))
1310, 11, 12sylancl 692 . . . . 5 (𝐷 ∈ (∞Met‘𝑋) → (𝑓 ∈ ((𝑋𝑌) ↑pm ℂ) ↔ (Fun 𝑓𝑓 ⊆ (ℂ × (𝑋𝑌)))))
14 elpmg 7733 . . . . . 6 ((𝑋 ∈ dom ∞Met ∧ ℂ ∈ V) → (𝑓 ∈ (𝑋pm ℂ) ↔ (Fun 𝑓𝑓 ⊆ (ℂ × 𝑋))))
158, 11, 14sylancl 692 . . . . 5 (𝐷 ∈ (∞Met‘𝑋) → (𝑓 ∈ (𝑋pm ℂ) ↔ (Fun 𝑓𝑓 ⊆ (ℂ × 𝑋))))
167, 13, 153imtr4d 281 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → (𝑓 ∈ ((𝑋𝑌) ↑pm ℂ) → 𝑓 ∈ (𝑋pm ℂ)))
17 uzid 11531 . . . . . . . . . 10 (𝑦 ∈ ℤ → 𝑦 ∈ (ℤ𝑦))
1817adantl 480 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) → 𝑦 ∈ (ℤ𝑦))
19 simp2 1054 . . . . . . . . . 10 ((𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥) → (𝑓𝑧) ∈ (𝑋𝑌))
2019ralimi 2932 . . . . . . . . 9 (∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥) → ∀𝑧 ∈ (ℤ𝑦)(𝑓𝑧) ∈ (𝑋𝑌))
21 fveq2 6085 . . . . . . . . . . 11 (𝑧 = 𝑦 → (𝑓𝑧) = (𝑓𝑦))
2221eleq1d 2668 . . . . . . . . . 10 (𝑧 = 𝑦 → ((𝑓𝑧) ∈ (𝑋𝑌) ↔ (𝑓𝑦) ∈ (𝑋𝑌)))
2322rspcva 3276 . . . . . . . . 9 ((𝑦 ∈ (ℤ𝑦) ∧ ∀𝑧 ∈ (ℤ𝑦)(𝑓𝑧) ∈ (𝑋𝑌)) → (𝑓𝑦) ∈ (𝑋𝑌))
2418, 20, 23syl2an 492 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ ∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥)) → (𝑓𝑦) ∈ (𝑋𝑌))
25 inss2 3792 . . . . . . . . . . . . . 14 (𝑋𝑌) ⊆ 𝑌
26 simpr 475 . . . . . . . . . . . . . 14 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓𝑦) ∈ (𝑋𝑌)) → (𝑓𝑦) ∈ (𝑋𝑌))
2725, 26sseldi 3562 . . . . . . . . . . . . 13 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓𝑦) ∈ (𝑋𝑌)) → (𝑓𝑦) ∈ 𝑌)
2825a1i 11 . . . . . . . . . . . . . . . . . . 19 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓𝑦) ∈ 𝑌) → (𝑋𝑌) ⊆ 𝑌)
2928sselda 3564 . . . . . . . . . . . . . . . . . 18 ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓𝑦) ∈ 𝑌) ∧ (𝑓𝑧) ∈ (𝑋𝑌)) → (𝑓𝑧) ∈ 𝑌)
30 simplr 787 . . . . . . . . . . . . . . . . . 18 ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓𝑦) ∈ 𝑌) ∧ (𝑓𝑧) ∈ (𝑋𝑌)) → (𝑓𝑦) ∈ 𝑌)
3129, 30ovresd 6674 . . . . . . . . . . . . . . . . 17 ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓𝑦) ∈ 𝑌) ∧ (𝑓𝑧) ∈ (𝑋𝑌)) → ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) = ((𝑓𝑧)𝐷(𝑓𝑦)))
3231breq1d 4584 . . . . . . . . . . . . . . . 16 ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓𝑦) ∈ 𝑌) ∧ (𝑓𝑧) ∈ (𝑋𝑌)) → (((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥 ↔ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥))
3332biimpd 217 . . . . . . . . . . . . . . 15 ((((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓𝑦) ∈ 𝑌) ∧ (𝑓𝑧) ∈ (𝑋𝑌)) → (((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥 → ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥))
3433imdistanda 724 . . . . . . . . . . . . . 14 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓𝑦) ∈ 𝑌) → (((𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥) → ((𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥)))
351a1i 11 . . . . . . . . . . . . . . . 16 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓𝑦) ∈ 𝑌) → (𝑋𝑌) ⊆ 𝑋)
3635sseld 3563 . . . . . . . . . . . . . . 15 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓𝑦) ∈ 𝑌) → ((𝑓𝑧) ∈ (𝑋𝑌) → (𝑓𝑧) ∈ 𝑋))
3736anim1d 585 . . . . . . . . . . . . . 14 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓𝑦) ∈ 𝑌) → (((𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥) → ((𝑓𝑧) ∈ 𝑋 ∧ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥)))
3834, 37syld 45 . . . . . . . . . . . . 13 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓𝑦) ∈ 𝑌) → (((𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥) → ((𝑓𝑧) ∈ 𝑋 ∧ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥)))
3927, 38syldan 485 . . . . . . . . . . . 12 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓𝑦) ∈ (𝑋𝑌)) → (((𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥) → ((𝑓𝑧) ∈ 𝑋 ∧ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥)))
4039anim2d 586 . . . . . . . . . . 11 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓𝑦) ∈ (𝑋𝑌)) → ((𝑧 ∈ dom 𝑓 ∧ ((𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥)) → (𝑧 ∈ dom 𝑓 ∧ ((𝑓𝑧) ∈ 𝑋 ∧ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥))))
41 3anass 1034 . . . . . . . . . . 11 ((𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥) ↔ (𝑧 ∈ dom 𝑓 ∧ ((𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥)))
42 3anass 1034 . . . . . . . . . . 11 ((𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ 𝑋 ∧ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥) ↔ (𝑧 ∈ dom 𝑓 ∧ ((𝑓𝑧) ∈ 𝑋 ∧ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥)))
4340, 41, 423imtr4g 283 . . . . . . . . . 10 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓𝑦) ∈ (𝑋𝑌)) → ((𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥) → (𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ 𝑋 ∧ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥)))
4443ralimdv 2942 . . . . . . . . 9 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ (𝑓𝑦) ∈ (𝑋𝑌)) → (∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥) → ∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ 𝑋 ∧ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥)))
4544impancom 454 . . . . . . . 8 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ ∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥)) → ((𝑓𝑦) ∈ (𝑋𝑌) → ∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ 𝑋 ∧ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥)))
4624, 45mpd 15 . . . . . . 7 (((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) ∧ ∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥)) → ∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ 𝑋 ∧ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥))
4746ex 448 . . . . . 6 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑦 ∈ ℤ) → (∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥) → ∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ 𝑋 ∧ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥)))
4847reximdva 2996 . . . . 5 (𝐷 ∈ (∞Met‘𝑋) → (∃𝑦 ∈ ℤ ∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥) → ∃𝑦 ∈ ℤ ∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ 𝑋 ∧ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥)))
4948ralimdv 2942 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → (∀𝑥 ∈ ℝ+𝑦 ∈ ℤ ∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥) → ∀𝑥 ∈ ℝ+𝑦 ∈ ℤ ∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ 𝑋 ∧ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥)))
5016, 49anim12d 583 . . 3 (𝐷 ∈ (∞Met‘𝑋) → ((𝑓 ∈ ((𝑋𝑌) ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℤ ∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥)) → (𝑓 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℤ ∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ 𝑋 ∧ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥))))
51 xmetres 21917 . . . 4 (𝐷 ∈ (∞Met‘𝑋) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘(𝑋𝑌)))
52 iscau2 22798 . . . 4 ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘(𝑋𝑌)) → (𝑓 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))) ↔ (𝑓 ∈ ((𝑋𝑌) ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℤ ∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥))))
5351, 52syl 17 . . 3 (𝐷 ∈ (∞Met‘𝑋) → (𝑓 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))) ↔ (𝑓 ∈ ((𝑋𝑌) ↑pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℤ ∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ (𝑋𝑌) ∧ ((𝑓𝑧)(𝐷 ↾ (𝑌 × 𝑌))(𝑓𝑦)) < 𝑥))))
54 iscau2 22798 . . 3 (𝐷 ∈ (∞Met‘𝑋) → (𝑓 ∈ (Cau‘𝐷) ↔ (𝑓 ∈ (𝑋pm ℂ) ∧ ∀𝑥 ∈ ℝ+𝑦 ∈ ℤ ∀𝑧 ∈ (ℤ𝑦)(𝑧 ∈ dom 𝑓 ∧ (𝑓𝑧) ∈ 𝑋 ∧ ((𝑓𝑧)𝐷(𝑓𝑦)) < 𝑥))))
5550, 53, 543imtr4d 281 . 2 (𝐷 ∈ (∞Met‘𝑋) → (𝑓 ∈ (Cau‘(𝐷 ↾ (𝑌 × 𝑌))) → 𝑓 ∈ (Cau‘𝐷)))
5655ssrdv 3570 1 (𝐷 ∈ (∞Met‘𝑋) → (Cau‘(𝐷 ↾ (𝑌 × 𝑌))) ⊆ (Cau‘𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  w3a 1030  wcel 1976  wral 2892  wrex 2893  Vcvv 3169  cin 3535  wss 3536   class class class wbr 4574   × cxp 5023  dom cdm 5025  cres 5027  Fun wfun 5781  cfv 5787  (class class class)co 6524  pm cpm 7719  cc 9787   < clt 9927  cz 11207  cuz 11516  +crp 11661  ∞Metcxmt 19495  Caucca 22774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586  ax-sep 4700  ax-nul 4709  ax-pow 4761  ax-pr 4825  ax-un 6821  ax-cnex 9845  ax-resscn 9846  ax-1cn 9847  ax-icn 9848  ax-addcl 9849  ax-addrcl 9850  ax-mulcl 9851  ax-mulrcl 9852  ax-mulcom 9853  ax-addass 9854  ax-mulass 9855  ax-distr 9856  ax-i2m1 9857  ax-1ne0 9858  ax-1rid 9859  ax-rnegex 9860  ax-rrecex 9861  ax-cnre 9862  ax-pre-lttri 9863  ax-pre-lttrn 9864  ax-pre-ltadd 9865
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2458  df-mo 2459  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-ne 2778  df-nel 2779  df-ral 2897  df-rex 2898  df-rab 2901  df-v 3171  df-sbc 3399  df-csb 3496  df-dif 3539  df-un 3541  df-in 3543  df-ss 3550  df-nul 3871  df-if 4033  df-pw 4106  df-sn 4122  df-pr 4124  df-op 4128  df-uni 4364  df-iun 4448  df-br 4575  df-opab 4635  df-mpt 4636  df-id 4940  df-po 4946  df-so 4947  df-xp 5031  df-rel 5032  df-cnv 5033  df-co 5034  df-dm 5035  df-rn 5036  df-res 5037  df-ima 5038  df-iota 5751  df-fun 5789  df-fn 5790  df-f 5791  df-f1 5792  df-fo 5793  df-f1o 5794  df-fv 5795  df-ov 6527  df-oprab 6528  df-mpt2 6529  df-1st 7033  df-2nd 7034  df-er 7603  df-map 7720  df-pm 7721  df-en 7816  df-dom 7817  df-sdom 7818  df-pnf 9929  df-mnf 9930  df-xr 9931  df-ltxr 9932  df-le 9933  df-neg 10117  df-z 11208  df-uz 11517  df-rp 11662  df-xadd 11776  df-psmet 19502  df-xmet 19503  df-bl 19505  df-cau 22777
This theorem is referenced by: (None)
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