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Theorem fzval 9107
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  NN_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  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N
)  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) } )
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 3796 . . . 4  |-  ( m  =  M  ->  (
m  <_  k  <->  M  <_  k ) )
21anbi1d 453 . . 3  |-  ( m  =  M  ->  (
( m  <_  k  /\  k  <_  n )  <-> 
( M  <_  k  /\  k  <_  n ) ) )
32rabbidv 2594 . 2  |-  ( m  =  M  ->  { k  e.  ZZ  |  ( m  <_  k  /\  k  <_  n ) }  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  n ) } )
4 breq2 3797 . . . 4  |-  ( n  =  N  ->  (
k  <_  n  <->  k  <_  N ) )
54anbi2d 452 . . 3  |-  ( n  =  N  ->  (
( M  <_  k  /\  k  <_  n )  <-> 
( M  <_  k  /\  k  <_  N ) ) )
65rabbidv 2594 . 2  |-  ( n  =  N  ->  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  n ) }  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) } )
7 df-fz 9106 . 2  |-  ...  =  ( m  e.  ZZ ,  n  e.  ZZ  |->  { k  e.  ZZ  |  ( m  <_ 
k  /\  k  <_  n ) } )
8 zex 8441 . . 3  |-  ZZ  e.  _V
98rabex 3930 . 2  |-  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) }  e.  _V
103, 6, 7, 9ovmpt2 5667 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N
)  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434   {crab 2353   class class class wbr 3793  (class class class)co 5543    <_ cle 7216   ZZcz 8432   ...cfz 9105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-setind 4288  ax-cnex 7129  ax-resscn 7130
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-iota 4897  df-fun 4934  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-neg 7349  df-z 8433  df-fz 9106
This theorem is referenced by:  fzval2  9108  elfz1  9110  fznlem  9136
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