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Theorem fzval 11670
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 4450 . . . 4  |-  ( m  =  M  ->  (
m  <_  k  <->  M  <_  k ) )
21anbi1d 704 . . 3  |-  ( m  =  M  ->  (
( m  <_  k  /\  k  <_  n )  <-> 
( M  <_  k  /\  k  <_  n ) ) )
32rabbidv 3105 . 2  |-  ( m  =  M  ->  { k  e.  ZZ  |  ( m  <_  k  /\  k  <_  n ) }  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  n ) } )
4 breq2 4451 . . . 4  |-  ( n  =  N  ->  (
k  <_  n  <->  k  <_  N ) )
54anbi2d 703 . . 3  |-  ( n  =  N  ->  (
( M  <_  k  /\  k  <_  n )  <-> 
( M  <_  k  /\  k  <_  N ) ) )
65rabbidv 3105 . 2  |-  ( n  =  N  ->  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  n ) }  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) } )
7 df-fz 11669 . 2  |-  ...  =  ( m  e.  ZZ ,  n  e.  ZZ  |->  { k  e.  ZZ  |  ( m  <_ 
k  /\  k  <_  n ) } )
8 zex 10869 . . 3  |-  ZZ  e.  _V
98rabex 4598 . 2  |-  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) }  e.  _V
103, 6, 7, 9ovmpt2 6420 1  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M ... N
)  =  { k  e.  ZZ  |  ( M  <_  k  /\  k  <_  N ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {crab 2818   class class class wbr 4447  (class class class)co 6282    <_ cle 9625   ZZcz 10860   ...cfz 11668
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-cnex 9544  ax-resscn 9545
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5549  df-fun 5588  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-neg 9804  df-z 10861  df-fz 11669
This theorem is referenced by:  fzval2  11671  elfz1  11673
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