Theorem uzval 9597

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 → (ℤ≥‘𝑁) = {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘})

Distinct variable group:   𝑘,𝑁

Proof of Theorem uzval

Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq1 4033	. . 3 ⊢ (𝑗 = 𝑁 → (𝑗 ≤ 𝑘 ↔ 𝑁 ≤ 𝑘))
2	1	rabbidv 2749	. 2 ⊢ (𝑗 = 𝑁 → {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} = {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘})
3		df-uz 9596	. 2 ⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘})
4		zex 9329	. . 3 ⊢ ℤ ∈ V
5	4	rabex 4174	. 2 ⊢ {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘} ∈ V
6	2, 3, 5	fvmpt 5635	1 ⊢ (𝑁 ∈ ℤ → (ℤ≥‘𝑁) = {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘})

Syntax hints:   → wi 4   = wceq 1364   ∈ wcel 2164  {crab 2476   class class class wbr 4030  ‘cfv 5255   ≤ cle 8057  ℤcz 9320  ℤ_≥cuz 9595

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-cnex 7965  ax-resscn 7966

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-iota 5216  df-fun 5257  df-fv 5263  df-ov 5922  df-neg 8195  df-z 9321  df-uz 9596

This theorem is referenced by:  eluz1  9599  nn0uz  9630  nnuz  9631  algfx  12193