Theorem bitsf 12509

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 12504	. 2 ⊢ bits = (𝑛 ∈ ℤ ↦ {𝑘 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑘)))})
2		nn0ex 9408	. . . 4 ⊢ ℕ0 ∈ V
3		ssrab2 3312	. . . 4 ⊢ {𝑘 ∈ ℕ0 ∣ ¬ 2 ∥ (⌊‘(𝑛 / (2↑𝑘)))} ⊆ ℕ0
4	2, 3	elpwi2 4248	. . 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 5799	1 ⊢ bits:ℤ⟶𝒫 ℕ0

Syntax hints:  ¬ wn 3   ∈ wcel 2202  {crab 2514  Vcvv 2802  𝒫 cpw 3652   class class class wbr 4088  ⟶wf 5322  ‘cfv 5326  (class class class)co 6018   / cdiv 8852  2c2 9194  ℕ₀cn0 9402  ℤcz 9479  ⌊cfl 10529  ↑cexp 10801   ∥ cdvds 12350  bitscbits 12503

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-i2m1 8137

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-inn 9144  df-n0 9403  df-bits 12504

This theorem is referenced by: (None)