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Theorem fzval 10784
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 4026 . . . 4  |-  ( m  =  M  ->  (
m  <_  k  <->  M  <_  k ) )
21anbi1d 685 . . 3  |-  ( m  =  M  ->  (
( m  <_  k  /\  k  <_  n )  <-> 
( M  <_  k  /\  k  <_  n ) ) )
32rabbidv 2780 . 2  |-  ( m  =  M  ->  { k  e.  ZZ  |  ( m  <_  k  /\  k  <_  n ) }  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  n ) } )
4 breq2 4027 . . . 4  |-  ( n  =  N  ->  (
k  <_  n  <->  k  <_  N ) )
54anbi2d 684 . . 3  |-  ( n  =  N  ->  (
( M  <_  k  /\  k  <_  n )  <-> 
( M  <_  k  /\  k  <_  N ) ) )
65rabbidv 2780 . 2  |-  ( n  =  N  ->  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  n ) }  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) } )
7 df-fz 10783 . 2  |-  ...  =  ( m  e.  ZZ ,  n  e.  ZZ  |->  { k  e.  ZZ  |  ( m  <_ 
k  /\  k  <_  n ) } )
8 zex 10033 . . 3  |-  ZZ  e.  _V
98rabex 4165 . 2  |-  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) }  e.  _V
103, 6, 7, 9ovmpt2 5983 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 358    = wceq 1623    e. wcel 1684   {crab 2547   class class class wbr 4023  (class class class)co 5858    <_ cle 8868   ZZcz 10024   ...cfz 10782
This theorem is referenced by:  fzval2  10785  elfz1  10787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-neg 9040  df-z 10025  df-fz 10783
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