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Theorem bitsf1ocnv 15004
 Description: The bits function restricted to nonnegative integers is a bijection from the integers to the finite sets of integers. It is in fact the inverse of the Ackermann bijection ackbijnn 14399. (Contributed by Mario Carneiro, 8-Sep-2016.)
Assertion
Ref Expression
bitsf1ocnv ((bits ↾ ℕ0):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin) ∧ (bits ↾ ℕ0) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑛𝑥 (2↑𝑛)))
Distinct variable group:   𝑥,𝑛

Proof of Theorem bitsf1ocnv
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 eqid 2610 . . . . . 6 (𝑘 ∈ ℕ0 ↦ (bits‘𝑘)) = (𝑘 ∈ ℕ0 ↦ (bits‘𝑘))
2 bitsss 14986 . . . . . . . . 9 (bits‘𝑘) ⊆ ℕ0
32a1i 11 . . . . . . . 8 (𝑘 ∈ ℕ0 → (bits‘𝑘) ⊆ ℕ0)
4 bitsfi 14997 . . . . . . . 8 (𝑘 ∈ ℕ0 → (bits‘𝑘) ∈ Fin)
5 elfpw 8151 . . . . . . . 8 ((bits‘𝑘) ∈ (𝒫 ℕ0 ∩ Fin) ↔ ((bits‘𝑘) ⊆ ℕ0 ∧ (bits‘𝑘) ∈ Fin))
63, 4, 5sylanbrc 695 . . . . . . 7 (𝑘 ∈ ℕ0 → (bits‘𝑘) ∈ (𝒫 ℕ0 ∩ Fin))
76adantl 481 . . . . . 6 ((⊤ ∧ 𝑘 ∈ ℕ0) → (bits‘𝑘) ∈ (𝒫 ℕ0 ∩ Fin))
8 elfpw 8151 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↔ (𝑥 ⊆ ℕ0𝑥 ∈ Fin))
98simprbi 479 . . . . . . . 8 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ∈ Fin)
10 2nn0 11186 . . . . . . . . . 10 2 ∈ ℕ0
1110a1i 11 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑛𝑥) → 2 ∈ ℕ0)
128simplbi 475 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 ⊆ ℕ0)
1312sselda 3568 . . . . . . . . 9 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑛𝑥) → 𝑛 ∈ ℕ0)
1411, 13nn0expcld 12893 . . . . . . . 8 ((𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ∧ 𝑛𝑥) → (2↑𝑛) ∈ ℕ0)
159, 14fsumnn0cl 14314 . . . . . . 7 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → Σ𝑛𝑥 (2↑𝑛) ∈ ℕ0)
1615adantl 481 . . . . . 6 ((⊤ ∧ 𝑥 ∈ (𝒫 ℕ0 ∩ Fin)) → Σ𝑛𝑥 (2↑𝑛) ∈ ℕ0)
17 bitsinv2 15003 . . . . . . . . . 10 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → (bits‘Σ𝑛𝑥 (2↑𝑛)) = 𝑥)
1817eqcomd 2616 . . . . . . . . 9 (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) → 𝑥 = (bits‘Σ𝑛𝑥 (2↑𝑛)))
1918ad2antll 761 . . . . . . . 8 ((⊤ ∧ (𝑘 ∈ ℕ0𝑥 ∈ (𝒫 ℕ0 ∩ Fin))) → 𝑥 = (bits‘Σ𝑛𝑥 (2↑𝑛)))
20 fveq2 6103 . . . . . . . . 9 (𝑘 = Σ𝑛𝑥 (2↑𝑛) → (bits‘𝑘) = (bits‘Σ𝑛𝑥 (2↑𝑛)))
2120eqeq2d 2620 . . . . . . . 8 (𝑘 = Σ𝑛𝑥 (2↑𝑛) → (𝑥 = (bits‘𝑘) ↔ 𝑥 = (bits‘Σ𝑛𝑥 (2↑𝑛))))
2219, 21syl5ibrcom 236 . . . . . . 7 ((⊤ ∧ (𝑘 ∈ ℕ0𝑥 ∈ (𝒫 ℕ0 ∩ Fin))) → (𝑘 = Σ𝑛𝑥 (2↑𝑛) → 𝑥 = (bits‘𝑘)))
23 bitsinv1 15002 . . . . . . . . . 10 (𝑘 ∈ ℕ0 → Σ𝑛 ∈ (bits‘𝑘)(2↑𝑛) = 𝑘)
2423eqcomd 2616 . . . . . . . . 9 (𝑘 ∈ ℕ0𝑘 = Σ𝑛 ∈ (bits‘𝑘)(2↑𝑛))
2524ad2antrl 760 . . . . . . . 8 ((⊤ ∧ (𝑘 ∈ ℕ0𝑥 ∈ (𝒫 ℕ0 ∩ Fin))) → 𝑘 = Σ𝑛 ∈ (bits‘𝑘)(2↑𝑛))
26 sumeq1 14267 . . . . . . . . 9 (𝑥 = (bits‘𝑘) → Σ𝑛𝑥 (2↑𝑛) = Σ𝑛 ∈ (bits‘𝑘)(2↑𝑛))
2726eqeq2d 2620 . . . . . . . 8 (𝑥 = (bits‘𝑘) → (𝑘 = Σ𝑛𝑥 (2↑𝑛) ↔ 𝑘 = Σ𝑛 ∈ (bits‘𝑘)(2↑𝑛)))
2825, 27syl5ibrcom 236 . . . . . . 7 ((⊤ ∧ (𝑘 ∈ ℕ0𝑥 ∈ (𝒫 ℕ0 ∩ Fin))) → (𝑥 = (bits‘𝑘) → 𝑘 = Σ𝑛𝑥 (2↑𝑛)))
2922, 28impbid 201 . . . . . 6 ((⊤ ∧ (𝑘 ∈ ℕ0𝑥 ∈ (𝒫 ℕ0 ∩ Fin))) → (𝑘 = Σ𝑛𝑥 (2↑𝑛) ↔ 𝑥 = (bits‘𝑘)))
301, 7, 16, 29f1ocnv2d 6784 . . . . 5 (⊤ → ((𝑘 ∈ ℕ0 ↦ (bits‘𝑘)):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin) ∧ (𝑘 ∈ ℕ0 ↦ (bits‘𝑘)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑛𝑥 (2↑𝑛))))
3130simpld 474 . . . 4 (⊤ → (𝑘 ∈ ℕ0 ↦ (bits‘𝑘)):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin))
32 bitsf 14987 . . . . . . . . 9 bits:ℤ⟶𝒫 ℕ0
3332a1i 11 . . . . . . . 8 (⊤ → bits:ℤ⟶𝒫 ℕ0)
3433feqmptd 6159 . . . . . . 7 (⊤ → bits = (𝑘 ∈ ℤ ↦ (bits‘𝑘)))
3534reseq1d 5316 . . . . . 6 (⊤ → (bits ↾ ℕ0) = ((𝑘 ∈ ℤ ↦ (bits‘𝑘)) ↾ ℕ0))
36 nn0ssz 11275 . . . . . . 7 0 ⊆ ℤ
37 resmpt 5369 . . . . . . 7 (ℕ0 ⊆ ℤ → ((𝑘 ∈ ℤ ↦ (bits‘𝑘)) ↾ ℕ0) = (𝑘 ∈ ℕ0 ↦ (bits‘𝑘)))
3836, 37ax-mp 5 . . . . . 6 ((𝑘 ∈ ℤ ↦ (bits‘𝑘)) ↾ ℕ0) = (𝑘 ∈ ℕ0 ↦ (bits‘𝑘))
3935, 38syl6eq 2660 . . . . 5 (⊤ → (bits ↾ ℕ0) = (𝑘 ∈ ℕ0 ↦ (bits‘𝑘)))
40 f1oeq1 6040 . . . . 5 ((bits ↾ ℕ0) = (𝑘 ∈ ℕ0 ↦ (bits‘𝑘)) → ((bits ↾ ℕ0):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin) ↔ (𝑘 ∈ ℕ0 ↦ (bits‘𝑘)):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin)))
4139, 40syl 17 . . . 4 (⊤ → ((bits ↾ ℕ0):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin) ↔ (𝑘 ∈ ℕ0 ↦ (bits‘𝑘)):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin)))
4231, 41mpbird 246 . . 3 (⊤ → (bits ↾ ℕ0):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin))
4339cnveqd 5220 . . . 4 (⊤ → (bits ↾ ℕ0) = (𝑘 ∈ ℕ0 ↦ (bits‘𝑘)))
4430simprd 478 . . . 4 (⊤ → (𝑘 ∈ ℕ0 ↦ (bits‘𝑘)) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑛𝑥 (2↑𝑛)))
4543, 44eqtrd 2644 . . 3 (⊤ → (bits ↾ ℕ0) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑛𝑥 (2↑𝑛)))
4642, 45jca 553 . 2 (⊤ → ((bits ↾ ℕ0):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin) ∧ (bits ↾ ℕ0) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑛𝑥 (2↑𝑛))))
4746trud 1484 1 ((bits ↾ ℕ0):ℕ01-1-onto→(𝒫 ℕ0 ∩ Fin) ∧ (bits ↾ ℕ0) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑛𝑥 (2↑𝑛)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ⊤wtru 1476   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108   ↦ cmpt 4643  ◡ccnv 5037   ↾ cres 5040  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  2c2 10947  ℕ0cn0 11169  ℤcz 11254  ↑cexp 12722  Σcsu 14264  bitscbits 14979 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-disj 4554  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-sup 8231  df-inf 8232  df-oi 8298  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-div 10564  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-xnn0 11241  df-z 11255  df-uz 11564  df-rp 11709  df-fz 12198  df-fzo 12335  df-fl 12455  df-mod 12531  df-seq 12664  df-exp 12723  df-hash 12980  df-cj 13687  df-re 13688  df-im 13689  df-sqrt 13823  df-abs 13824  df-clim 14067  df-sum 14265  df-dvds 14822  df-bits 14982 This theorem is referenced by:  bitsf1o  15005  bitsinv  15008
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