Users' Mathboxes Mathbox for Jeff Madsen < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sstotbnd3 Structured version   Visualization version   GIF version

Theorem sstotbnd3 33234
Description: Use a net that is not necessarily finite, but for which only finitely many balls meet the subset. (Contributed by Mario Carneiro, 14-Sep-2015.)
Hypothesis
Ref Expression
sstotbnd.2 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))
Assertion
Ref Expression
sstotbnd3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ 𝒫 𝑋(𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
Distinct variable groups:   𝑣,𝑑,𝑥,𝑀   𝑋,𝑑,𝑣,𝑥   𝑁,𝑑,𝑣,𝑥   𝑌,𝑑,𝑣,𝑥

Proof of Theorem sstotbnd3
Dummy variables 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sstotbnd.2 . . . 4 𝑁 = (𝑀 ↾ (𝑌 × 𝑌))
21sstotbnd2 33232 . . 3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑)))
3 elin 3779 . . . . . . . . 9 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑣 ∈ 𝒫 𝑋𝑣 ∈ Fin))
4 rabfi 8136 . . . . . . . . . 10 (𝑣 ∈ Fin → {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)
54anim2i 592 . . . . . . . . 9 ((𝑣 ∈ 𝒫 𝑋𝑣 ∈ Fin) → (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
63, 5sylbi 207 . . . . . . . 8 (𝑣 ∈ (𝒫 𝑋 ∩ Fin) → (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
76anim2i 592 . . . . . . 7 ((𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑣 ∈ (𝒫 𝑋 ∩ Fin)) → (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
87ancoms 469 . . . . . 6 ((𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑)) → (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
9 an12 837 . . . . . 6 ((𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ (𝑣 ∈ 𝒫 𝑋 ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) ↔ (𝑣 ∈ 𝒫 𝑋 ∧ (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
108, 9sylib 208 . . . . 5 ((𝑣 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑)) → (𝑣 ∈ 𝒫 𝑋 ∧ (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
1110reximi2 3005 . . . 4 (∃𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) → ∃𝑣 ∈ 𝒫 𝑋(𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
1211ralimi 2947 . . 3 (∀𝑑 ∈ ℝ+𝑣 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) → ∀𝑑 ∈ ℝ+𝑣 ∈ 𝒫 𝑋(𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
132, 12syl6bi 243 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) → ∀𝑑 ∈ ℝ+𝑣 ∈ 𝒫 𝑋(𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
14 ssrab2 3671 . . . . . . . . 9 {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ⊆ 𝑣
15 elpwi 4145 . . . . . . . . . 10 (𝑣 ∈ 𝒫 𝑋𝑣𝑋)
1615ad2antlr 762 . . . . . . . . 9 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → 𝑣𝑋)
1714, 16syl5ss 3598 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ⊆ 𝑋)
18 simprr 795 . . . . . . . 8 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)
19 elfpw 8219 . . . . . . . 8 ({𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ (𝒫 𝑋 ∩ Fin) ↔ ({𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ⊆ 𝑋 ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin))
2017, 18, 19sylanbrc 697 . . . . . . 7 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ (𝒫 𝑋 ∩ Fin))
21 ssel2 3582 . . . . . . . . . . . . 13 ((𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑧𝑌) → 𝑧 𝑥𝑣 (𝑥(ball‘𝑀)𝑑))
22 eliun 4495 . . . . . . . . . . . . 13 (𝑧 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ↔ ∃𝑥𝑣 𝑧 ∈ (𝑥(ball‘𝑀)𝑑))
2321, 22sylib 208 . . . . . . . . . . . 12 ((𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑧𝑌) → ∃𝑥𝑣 𝑧 ∈ (𝑥(ball‘𝑀)𝑑))
24 inelcm 4009 . . . . . . . . . . . . . . . 16 ((𝑧 ∈ (𝑥(ball‘𝑀)𝑑) ∧ 𝑧𝑌) → ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅)
2524expcom 451 . . . . . . . . . . . . . . 15 (𝑧𝑌 → (𝑧 ∈ (𝑥(ball‘𝑀)𝑑) → ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅))
2625ancrd 576 . . . . . . . . . . . . . 14 (𝑧𝑌 → (𝑧 ∈ (𝑥(ball‘𝑀)𝑑) → (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑))))
2726reximdv 3011 . . . . . . . . . . . . 13 (𝑧𝑌 → (∃𝑥𝑣 𝑧 ∈ (𝑥(ball‘𝑀)𝑑) → ∃𝑥𝑣 (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑))))
2827impcom 446 . . . . . . . . . . . 12 ((∃𝑥𝑣 𝑧 ∈ (𝑥(ball‘𝑀)𝑑) ∧ 𝑧𝑌) → ∃𝑥𝑣 (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
2923, 28sylancom 700 . . . . . . . . . . 11 ((𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑧𝑌) → ∃𝑥𝑣 (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
30 eliun 4495 . . . . . . . . . . . 12 (𝑧 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑) ↔ ∃𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅}𝑧 ∈ (𝑦(ball‘𝑀)𝑑))
31 oveq1 6617 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑦(ball‘𝑀)𝑑) = (𝑥(ball‘𝑀)𝑑))
3231eleq2d 2684 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (𝑧 ∈ (𝑦(ball‘𝑀)𝑑) ↔ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
3332rexrab2 3360 . . . . . . . . . . . 12 (∃𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅}𝑧 ∈ (𝑦(ball‘𝑀)𝑑) ↔ ∃𝑥𝑣 (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
3430, 33bitri 264 . . . . . . . . . . 11 (𝑧 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑) ↔ ∃𝑥𝑣 (((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅ ∧ 𝑧 ∈ (𝑥(ball‘𝑀)𝑑)))
3529, 34sylibr 224 . . . . . . . . . 10 ((𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ 𝑧𝑌) → 𝑧 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑))
3635ex 450 . . . . . . . . 9 (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) → (𝑧𝑌𝑧 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑)))
3736ssrdv 3593 . . . . . . . 8 (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) → 𝑌 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑))
3837ad2antrl 763 . . . . . . 7 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → 𝑌 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑))
39 iuneq1 4505 . . . . . . . . 9 (𝑤 = {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} → 𝑦𝑤 (𝑦(ball‘𝑀)𝑑) = 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑))
4039sseq2d 3617 . . . . . . . 8 (𝑤 = {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} → (𝑌 𝑦𝑤 (𝑦(ball‘𝑀)𝑑) ↔ 𝑌 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑)))
4140rspcev 3298 . . . . . . 7 (({𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑌 𝑦 ∈ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} (𝑦(ball‘𝑀)𝑑)) → ∃𝑤 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑦𝑤 (𝑦(ball‘𝑀)𝑑))
4220, 38, 41syl2anc 692 . . . . . 6 ((((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) ∧ (𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)) → ∃𝑤 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑦𝑤 (𝑦(ball‘𝑀)𝑑))
4342ex 450 . . . . 5 (((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) ∧ 𝑣 ∈ 𝒫 𝑋) → ((𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin) → ∃𝑤 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑦𝑤 (𝑦(ball‘𝑀)𝑑)))
4443rexlimdva 3025 . . . 4 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (∃𝑣 ∈ 𝒫 𝑋(𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin) → ∃𝑤 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑦𝑤 (𝑦(ball‘𝑀)𝑑)))
4544ralimdv 2958 . . 3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (∀𝑑 ∈ ℝ+𝑣 ∈ 𝒫 𝑋(𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin) → ∀𝑑 ∈ ℝ+𝑤 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑦𝑤 (𝑦(ball‘𝑀)𝑑)))
461sstotbnd2 33232 . . 3 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑤 ∈ (𝒫 𝑋 ∩ Fin)𝑌 𝑦𝑤 (𝑦(ball‘𝑀)𝑑)))
4745, 46sylibrd 249 . 2 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (∀𝑑 ∈ ℝ+𝑣 ∈ 𝒫 𝑋(𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin) → 𝑁 ∈ (TotBnd‘𝑌)))
4813, 47impbid 202 1 ((𝑀 ∈ (Met‘𝑋) ∧ 𝑌𝑋) → (𝑁 ∈ (TotBnd‘𝑌) ↔ ∀𝑑 ∈ ℝ+𝑣 ∈ 𝒫 𝑋(𝑌 𝑥𝑣 (𝑥(ball‘𝑀)𝑑) ∧ {𝑥𝑣 ∣ ((𝑥(ball‘𝑀)𝑑) ∩ 𝑌) ≠ ∅} ∈ Fin)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wne 2790  wral 2907  wrex 2908  {crab 2911  cin 3558  wss 3559  c0 3896  𝒫 cpw 4135   ciun 4490   × cxp 5077  cres 5081  cfv 5852  (class class class)co 6610  Fincfn 7906  +crp 11783  Metcme 19660  ballcbl 19661  TotBndctotbnd 33224
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-oadd 7516  df-er 7694  df-map 7811  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-div 10636  df-2 11030  df-rp 11784  df-xneg 11897  df-xadd 11898  df-xmul 11899  df-psmet 19666  df-xmet 19667  df-met 19668  df-bl 19669  df-totbnd 33226
This theorem is referenced by:  cntotbnd  33254
  Copyright terms: Public domain W3C validator