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Theorem fzval 12366
Description: The value of a finite set of sequential integers. E.g., 2...5 means the set {2, 3, 4, 5}. A special case of this definition (starting at 1) appears as Definition 11-2.1 of [Gleason] p. 141, where _k means our 1...𝑘; he calls these sets segments of the integers. (Contributed by NM, 6-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
fzval ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)})
Distinct variable groups:   𝑘,𝑀   𝑘,𝑁

Proof of Theorem fzval
Dummy variables 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 4688 . . . 4 (𝑚 = 𝑀 → (𝑚𝑘𝑀𝑘))
21anbi1d 741 . . 3 (𝑚 = 𝑀 → ((𝑚𝑘𝑘𝑛) ↔ (𝑀𝑘𝑘𝑛)))
32rabbidv 3220 . 2 (𝑚 = 𝑀 → {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑛)})
4 breq2 4689 . . . 4 (𝑛 = 𝑁 → (𝑘𝑛𝑘𝑁))
54anbi2d 740 . . 3 (𝑛 = 𝑁 → ((𝑀𝑘𝑘𝑛) ↔ (𝑀𝑘𝑘𝑁)))
65rabbidv 3220 . 2 (𝑛 = 𝑁 → {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)})
7 df-fz 12365 . 2 ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)})
8 zex 11424 . . 3 ℤ ∈ V
98rabex 4845 . 2 {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)} ∈ V
103, 6, 7, 9ovmpt2 6838 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  {crab 2945   class class class wbr 4685  (class class class)co 6690  cle 10113  cz 11415  ...cfz 12364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-cnex 10030  ax-resscn 10031
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-neg 10307  df-z 11416  df-fz 12365
This theorem is referenced by:  fzval2  12367  elfz1  12369
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