MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fzval Structured version   Visualization version   GIF version

Theorem fzval 13526
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 5155 . . . 4 (𝑚 = 𝑀 → (𝑚𝑘𝑀𝑘))
21anbi1d 629 . . 3 (𝑚 = 𝑀 → ((𝑚𝑘𝑘𝑛) ↔ (𝑀𝑘𝑘𝑛)))
32rabbidv 3438 . 2 (𝑚 = 𝑀 → {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑛)})
4 breq2 5156 . . . 4 (𝑛 = 𝑁 → (𝑘𝑛𝑘𝑁))
54anbi2d 628 . . 3 (𝑛 = 𝑁 → ((𝑀𝑘𝑘𝑛) ↔ (𝑀𝑘𝑘𝑁)))
65rabbidv 3438 . 2 (𝑛 = 𝑁 → {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)})
7 df-fz 13525 . 2 ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)})
8 zex 12605 . . 3 ℤ ∈ V
98rabex 5338 . 2 {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)} ∈ V
103, 6, 7, 9ovmpo 7587 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wcel 2098  {crab 3430   class class class wbr 5152  (class class class)co 7426  cle 11287  cz 12596  ...cfz 13524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-cnex 11202  ax-resscn 11203
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-neg 11485  df-z 12597  df-fz 13525
This theorem is referenced by:  fzval2  13527  elfz1  13529
  Copyright terms: Public domain W3C validator