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Theorem fzval 8977
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 3794 . . . 4 (𝑚 = 𝑀 → (𝑚𝑘𝑀𝑘))
21anbi1d 446 . . 3 (𝑚 = 𝑀 → ((𝑚𝑘𝑘𝑛) ↔ (𝑀𝑘𝑘𝑛)))
32rabbidv 2566 . 2 (𝑚 = 𝑀 → {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑛)})
4 breq2 3795 . . . 4 (𝑛 = 𝑁 → (𝑘𝑛𝑘𝑁))
54anbi2d 445 . . 3 (𝑛 = 𝑁 → ((𝑀𝑘𝑘𝑛) ↔ (𝑀𝑘𝑘𝑁)))
65rabbidv 2566 . 2 (𝑛 = 𝑁 → {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑛)} = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)})
7 df-fz 8976 . 2 ... = (𝑚 ∈ ℤ, 𝑛 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ (𝑚𝑘𝑘𝑛)})
8 zex 8310 . . 3 ℤ ∈ V
98rabex 3928 . 2 {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)} ∈ V
103, 6, 7, 9ovmpt2 5663 1 ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = {𝑘 ∈ ℤ ∣ (𝑀𝑘𝑘𝑁)})
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  {crab 2327   class class class wbr 3791  (class class class)co 5539  cle 7119  cz 8301  ...cfz 8975
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-setind 4289  ax-cnex 7032  ax-resscn 7033
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-iota 4894  df-fun 4931  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-neg 7247  df-z 8302  df-fz 8976
This theorem is referenced by:  fzval2  8978  elfz1  8980  fznlem  9006
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