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Theorem fzval 9909
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  NNk 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 3968 . . . 4  |-  ( m  =  M  ->  (
m  <_  k  <->  M  <_  k ) )
21anbi1d 461 . . 3  |-  ( m  =  M  ->  (
( m  <_  k  /\  k  <_  n )  <-> 
( M  <_  k  /\  k  <_  n ) ) )
32rabbidv 2701 . 2  |-  ( m  =  M  ->  { k  e.  ZZ  |  ( m  <_  k  /\  k  <_  n ) }  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  n ) } )
4 breq2 3969 . . . 4  |-  ( n  =  N  ->  (
k  <_  n  <->  k  <_  N ) )
54anbi2d 460 . . 3  |-  ( n  =  N  ->  (
( M  <_  k  /\  k  <_  n )  <-> 
( M  <_  k  /\  k  <_  N ) ) )
65rabbidv 2701 . 2  |-  ( n  =  N  ->  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  n ) }  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) } )
7 df-fz 9908 . 2  |-  ...  =  ( m  e.  ZZ ,  n  e.  ZZ  |->  { k  e.  ZZ  |  ( m  <_ 
k  /\  k  <_  n ) } )
8 zex 9171 . . 3  |-  ZZ  e.  _V
98rabex 4108 . 2  |-  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) }  e.  _V
103, 6, 7, 9ovmpo 5953 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 103    = wceq 1335    e. wcel 2128   {crab 2439   class class class wbr 3965  (class class class)co 5821    <_ cle 7908   ZZcz 9162   ...cfz 9907
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-setind 4495  ax-cnex 7818  ax-resscn 7819
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-xp 4591  df-rel 4592  df-cnv 4593  df-co 4594  df-dm 4595  df-iota 5134  df-fun 5171  df-fv 5177  df-ov 5824  df-oprab 5825  df-mpo 5826  df-neg 8044  df-z 9163  df-fz 9908
This theorem is referenced by:  fzval2  9910  elfz1  9912  fznlem  9938
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