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Theorem fzval 12535
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 4812 . . . 4 (𝑚 = 𝑀 → (𝑚𝑘𝑀𝑘))
21anbi1d 623 . . 3 (𝑚 = 𝑀 → ((𝑚𝑘𝑘𝑛) ↔ (𝑀𝑘𝑘𝑛)))
32rabbidv 3338 . 2 (𝑚 = 𝑀 → {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑛)})
4 breq2 4813 . . . 4 (𝑛 = 𝑁 → (𝑘𝑛𝑘𝑁))
54anbi2d 622 . . 3 (𝑛 = 𝑁 → ((𝑀𝑘𝑘𝑛) ↔ (𝑀𝑘𝑘𝑁)))
65rabbidv 3338 . 2 (𝑛 = 𝑁 → {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)})
7 df-fz 12534 . 2 ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)})
8 zex 11633 . . 3 ℤ ∈ V
98rabex 4973 . 2 {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)} ∈ V
103, 6, 7, 9ovmpt2 6994 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1652  wcel 2155  {crab 3059   class class class wbr 4809  (class class class)co 6842  cle 10329  cz 11624  ...cfz 12533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-cnex 10245  ax-resscn 10246
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-iota 6031  df-fun 6070  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-neg 10523  df-z 11625  df-fz 12534
This theorem is referenced by:  fzval2  12536  elfz1  12538
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