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Theorem fzval 12748
 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 ℕk 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 4971 . . . 4 (𝑚 = 𝑀 → (𝑚𝑘𝑀𝑘))
21anbi1d 629 . . 3 (𝑚 = 𝑀 → ((𝑚𝑘𝑘𝑛) ↔ (𝑀𝑘𝑘𝑛)))
32rabbidv 3428 . 2 (𝑚 = 𝑀 → {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑛)})
4 breq2 4972 . . . 4 (𝑛 = 𝑁 → (𝑘𝑛𝑘𝑁))
54anbi2d 628 . . 3 (𝑛 = 𝑁 → ((𝑀𝑘𝑘𝑛) ↔ (𝑀𝑘𝑘𝑁)))
65rabbidv 3428 . 2 (𝑛 = 𝑁 → {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)})
7 df-fz 12747 . 2 ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)})
8 zex 11844 . . 3 ℤ ∈ V
98rabex 5133 . 2 {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)} ∈ V
103, 6, 7, 9ovmpo 7173 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396   = wceq 1525   ∈ wcel 2083  {crab 3111   class class class wbr 4968  (class class class)co 7023   ≤ cle 10529  ℤcz 11835  ...cfz 12746 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228  ax-cnex 10446  ax-resscn 10447 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-uni 4752  df-br 4969  df-opab 5031  df-id 5355  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-iota 6196  df-fun 6234  df-fv 6240  df-ov 7026  df-oprab 7027  df-mpo 7028  df-neg 10726  df-z 11836  df-fz 12747 This theorem is referenced by:  fzval2  12749  elfz1  12751
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