Theorem bitsf 12465

Description: The bits function is a function from integers to subsets of nonnegative integers. (Contributed by Mario Carneiro, 5-Sep-2016.)

Assertion

Ref	Expression
bitsf	⊢ bits:ℤ⟶𝒫 ℕ0

Proof of Theorem bitsf

Dummy variables 𝑘 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bits 12460	. 2 ⊢ bits = (𝑛 ∈ ℤ ↦ {𝑘 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑘)))})
2		nn0ex 9383	. . . 4 ⊢ ℕ0 ∈ V
3		ssrab2 3309	. . . 4 ⊢ {𝑘 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑘)))} ⊆ ℕ0
4	2, 3	elpwi2 4242	. . 3 ⊢ {𝑘 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑘)))} ∈ 𝒫 ℕ0
5	4	a1i 9	. 2 ⊢ (𝑛 ∈ ℤ → {𝑘 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑘)))} ∈ 𝒫 ℕ0)
6	1, 5	fmpti 5789	1 ⊢ bits:ℤ⟶𝒫 ℕ0

Syntax hints:  ¬ wn 3   ∈ wcel 2200  {crab 2512  Vcvv 2799  𝒫 cpw 3649   class class class wbr 4083  ⟶wf 5314  ‘cfv 5318  (class class class)co 6007   / cdiv 8827  2c2 9169  ℕ₀cn0 9377  ℤcz 9454  ⌊cfl 10496  ↑cexp 10768   ∥ cdvds 12306  bitscbits 12459

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8098  ax-resscn 8099  ax-1cn 8100  ax-1re 8101  ax-icn 8102  ax-addcl 8103  ax-addrcl 8104  ax-mulcl 8105  ax-i2m1 8112

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-inn 9119  df-n0 9378  df-bits 12460

This theorem is referenced by: (None)