Theorem uzval 8621

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 3788	. . 3 ⊢ (𝑗 = 𝑁 → (𝑗 ≤ 𝑘 ↔ 𝑁 ≤ 𝑘))
2	1	rabbidv 2593	. 2 ⊢ (𝑗 = 𝑁 → {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘} = {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘})
3		df-uz 8620	. 2 ⊢ ℤ≥ = (𝑗 ∈ ℤ ↦ {𝑘 ∈ ℤ ∣ 𝑗 ≤ 𝑘})
4		zex 8360	. . 3 ⊢ ℤ ∈ V
5	4	rabex 3922	. 2 ⊢ {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘} ∈ V
6	2, 3, 5	fvmpt 5270	1 ⊢ (𝑁 ∈ ℤ → (ℤ≥‘𝑁) = {𝑘 ∈ ℤ ∣ 𝑁 ≤ 𝑘})

Syntax hints:   → wi 4   = wceq 1284   ∈ wcel 1433  {crab 2352   class class class wbr 3785  ‘cfv 4922   ≤ cle 7154  ℤcz 8351  ℤ_≥cuz 8619

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-cnex 7067  ax-resscn 7068

This theorem depends on definitions:  df-bi 115  df-3or 920  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-rab 2357  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-iota 4887  df-fun 4924  df-fv 4930  df-ov 5535  df-neg 7282  df-z 8352  df-uz 8620

This theorem is referenced by:  eluz1  8623  nn0uz  8653  nnuz  8654  ialgfx  10434