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Theorem fzval 10800
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 4042 . . . 4  |-  ( m  =  M  ->  (
m  <_  k  <->  M  <_  k ) )
21anbi1d 685 . . 3  |-  ( m  =  M  ->  (
( m  <_  k  /\  k  <_  n )  <-> 
( M  <_  k  /\  k  <_  n ) ) )
32rabbidv 2793 . 2  |-  ( m  =  M  ->  { k  e.  ZZ  |  ( m  <_  k  /\  k  <_  n ) }  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  n ) } )
4 breq2 4043 . . . 4  |-  ( n  =  N  ->  (
k  <_  n  <->  k  <_  N ) )
54anbi2d 684 . . 3  |-  ( n  =  N  ->  (
( M  <_  k  /\  k  <_  n )  <-> 
( M  <_  k  /\  k  <_  N ) ) )
65rabbidv 2793 . 2  |-  ( n  =  N  ->  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  n ) }  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) } )
7 df-fz 10799 . 2  |-  ...  =  ( m  e.  ZZ ,  n  e.  ZZ  |->  { k  e.  ZZ  |  ( m  <_ 
k  /\  k  <_  n ) } )
8 zex 10049 . . 3  |-  ZZ  e.  _V
98rabex 4181 . 2  |-  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) }  e.  _V
103, 6, 7, 9ovmpt2 5999 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 1632    e. wcel 1696   {crab 2560   class class class wbr 4039  (class class class)co 5874    <_ cle 8884   ZZcz 10040   ...cfz 10798
This theorem is referenced by:  fzval2  10801  elfz1  10803
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-cnex 8809  ax-resscn 8810
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-neg 9056  df-z 10041  df-fz 10799
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