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Theorem uzval 9528
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 ∈ ℤ → (ℤ𝑁) = {𝑘 ∈ ℤ ∣ 𝑁𝑘})
Distinct variable group:   𝑘,𝑁

Proof of Theorem uzval
Dummy variable 𝑗 is distinct from all other variables.
StepHypRef Expression
1 breq1 4006 . . 3 (𝑗 = 𝑁 → (𝑗𝑘𝑁𝑘))
21rabbidv 2726 . 2 (𝑗 = 𝑁 → {𝑘 ∈ ℤ ∣ 𝑗𝑘} = {𝑘 ∈ ℤ ∣ 𝑁𝑘})
3 df-uz 9527 . 2 = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗𝑘})
4 zex 9260 . . 3 ℤ ∈ V
54rabex 4147 . 2 {𝑘 ∈ ℤ ∣ 𝑁𝑘} ∈ V
62, 3, 5fvmpt 5593 1 (𝑁 ∈ ℤ → (ℤ𝑁) = {𝑘 ∈ ℤ ∣ 𝑁𝑘})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  {crab 2459   class class class wbr 4003  cfv 5216  cle 7991  cz 9251  cuz 9526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-cnex 7901  ax-resscn 7902
This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-ov 5877  df-neg 8129  df-z 9252  df-uz 9527
This theorem is referenced by:  eluz1  9530  nn0uz  9560  nnuz  9561  algfx  12046
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