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Theorem uzval 12823
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 5142 . . 3 (𝑗 = 𝑁 → (𝑗𝑘𝑁𝑘))
21rabbidv 3432 . 2 (𝑗 = 𝑁 → {𝑘 ∈ ℤ ∣ 𝑗𝑘} = {𝑘 ∈ ℤ ∣ 𝑁𝑘})
3 df-uz 12822 . 2 = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗𝑘})
4 zex 12566 . . 3 ℤ ∈ V
54rabex 5323 . 2 {𝑘 ∈ ℤ ∣ 𝑁𝑘} ∈ V
62, 3, 5fvmpt 6989 1 (𝑁 ∈ ℤ → (ℤ𝑁) = {𝑘 ∈ ℤ ∣ 𝑁𝑘})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  {crab 3424   class class class wbr 5139  cfv 6534  cle 11248  cz 12557  cuz 12821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-cnex 11163  ax-resscn 11164
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-ov 7405  df-neg 11446  df-z 12558  df-uz 12822
This theorem is referenced by:  eluz1  12825  nn0uz  12863  nnuz  12864  algfx  16520
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