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Theorem uzval 8702
Description: The value of the upper integers function. (Contributed by NM, 5-Sep-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
uzval  |-  ( N  e.  ZZ  ->  ( ZZ>=
`  N )  =  { k  e.  ZZ  |  N  <_  k } )
Distinct variable group:    k, N

Proof of Theorem uzval
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 breq1 3796 . . 3  |-  ( j  =  N  ->  (
j  <_  k  <->  N  <_  k ) )
21rabbidv 2594 . 2  |-  ( j  =  N  ->  { k  e.  ZZ  |  j  <_  k }  =  { k  e.  ZZ  |  N  <_  k } )
3 df-uz 8701 . 2  |-  ZZ>=  =  ( j  e.  ZZ  |->  { k  e.  ZZ  | 
j  <_  k }
)
4 zex 8441 . . 3  |-  ZZ  e.  _V
54rabex 3930 . 2  |-  { k  e.  ZZ  |  N  <_  k }  e.  _V
62, 3, 5fvmpt 5281 1  |-  ( N  e.  ZZ  ->  ( ZZ>=
`  N )  =  { k  e.  ZZ  |  N  <_  k } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285    e. wcel 1434   {crab 2353   class class class wbr 3793   ` cfv 4932    <_ cle 7216   ZZcz 8432   ZZ>=cuz 8700
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-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-cnex 7129  ax-resscn 7130
This theorem depends on definitions:  df-bi 115  df-3or 921  df-3an 922  df-tru 1288  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-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  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-mpt 3849  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-neg 7349  df-z 8433  df-uz 8701
This theorem is referenced by:  eluz1  8704  nn0uz  8734  nnuz  8735  ialgfx  10578
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