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Theorem uzval 8570
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 3794 . . 3 (𝑗 = 𝑁 → (𝑗𝑘𝑁𝑘))
21rabbidv 2566 . 2 (𝑗 = 𝑁 → {𝑘 ∈ ℤ ∣ 𝑗𝑘} = {𝑘 ∈ ℤ ∣ 𝑁𝑘})
3 df-uz 8569 . 2 = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗𝑘})
4 zex 8310 . . 3 ℤ ∈ V
54rabex 3928 . 2 {𝑘 ∈ ℤ ∣ 𝑁𝑘} ∈ V
62, 3, 5fvmpt 5276 1 (𝑁 ∈ ℤ → (ℤ𝑁) = {𝑘 ∈ ℤ ∣ 𝑁𝑘})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1259  wcel 1409  {crab 2327   class class class wbr 3791  cfv 4929  cle 7119  cz 8301  cuz 8568
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-cnex 7032  ax-resscn 7033
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-iota 4894  df-fun 4931  df-fv 4937  df-ov 5542  df-neg 7247  df-z 8302  df-uz 8569
This theorem is referenced by:  eluz1  8572  nn0uz  8602  nnuz  8603  ialgfx  10260
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