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Theorem uzval 8251
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 3758 . . 3 (𝑗 = 𝑁 → (𝑗𝑘𝑁𝑘))
21rabbidv 2543 . 2 (𝑗 = 𝑁 → {𝑘 ℤ ∣ 𝑗𝑘} = {𝑘 ℤ ∣ 𝑁𝑘})
3 df-uz 8250 . 2 = (𝑗 ℤ ↦ {𝑘 ℤ ∣ 𝑗𝑘})
4 zex 8030 . . 3 V
54rabex 3892 . 2 {𝑘 ℤ ∣ 𝑁𝑘} V
62, 3, 5fvmpt 5192 1 (𝑁 ℤ → (ℤ𝑁) = {𝑘 ℤ ∣ 𝑁𝑘})
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  {crab 2304   class class class wbr 3755  cfv 4845  cle 6858  cz 8021  cuz 8249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-cnex 6774  ax-resscn 6775
This theorem depends on definitions:  df-bi 110  df-3or 885  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-ov 5458  df-neg 6982  df-z 8022  df-uz 8250
This theorem is referenced by:  eluz1  8253  nn0uz  8283  nnuz  8284
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