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Theorem uzval 9532
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 4008 . . 3 (𝑗 = 𝑁 → (𝑗𝑘𝑁𝑘))
21rabbidv 2728 . 2 (𝑗 = 𝑁 → {𝑘 ∈ ℤ ∣ 𝑗𝑘} = {𝑘 ∈ ℤ ∣ 𝑁𝑘})
3 df-uz 9531 . 2 = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗𝑘})
4 zex 9264 . . 3 ℤ ∈ V
54rabex 4149 . 2 {𝑘 ∈ ℤ ∣ 𝑁𝑘} ∈ V
62, 3, 5fvmpt 5595 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 4005  cfv 5218  cle 7995  cz 9255  cuz 9530
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 4123  ax-pow 4176  ax-pr 4211  ax-cnex 7904  ax-resscn 7905
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 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-ov 5880  df-neg 8133  df-z 9256  df-uz 9531
This theorem is referenced by:  eluz1  9534  nn0uz  9564  nnuz  9565  algfx  12054
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