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Mirrors > Home > MPE Home > Th. List > bitsf1o | Structured version Visualization version GIF version |
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 15163. (Contributed by Mario Carneiro, 8-Sep-2016.) |
Ref | Expression |
---|---|
bitsf1o | ⊢ (bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bitsf1ocnv 15771 | . 2 ⊢ ((bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin) ∧ ◡(bits ↾ ℕ0) = (𝑥 ∈ (𝒫 ℕ0 ∩ Fin) ↦ Σ𝑛 ∈ 𝑥 (2↑𝑛))) | |
2 | 1 | simpli 486 | 1 ⊢ (bits ↾ ℕ0):ℕ0–1-1-onto→(𝒫 ℕ0 ∩ Fin) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∩ cin 3918 𝒫 cpw 4520 ↦ cmpt 5127 ◡ccnv 5535 ↾ cres 5538 –1-1-onto→wf1o 6335 (class class class)co 7137 Fincfn 8490 2c2 11674 ℕ0cn0 11879 ↑cexp 13414 Σcsu 15022 bitscbits 15746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-inf2 9085 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 ax-pre-sup 10596 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rmo 3141 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-int 4858 df-iun 4902 df-disj 5013 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-se 5496 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-isom 6345 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-1st 7670 df-2nd 7671 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-1o 8083 df-2o 8084 df-oadd 8087 df-er 8270 df-map 8389 df-pm 8390 df-en 8491 df-dom 8492 df-sdom 8493 df-fin 8494 df-sup 8887 df-inf 8888 df-oi 8955 df-dju 9311 df-card 9349 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-div 11279 df-nn 11620 df-2 11682 df-3 11683 df-n0 11880 df-xnn0 11950 df-z 11964 df-uz 12226 df-rp 12372 df-fz 12878 df-fzo 13019 df-fl 13147 df-mod 13223 df-seq 13355 df-exp 13415 df-hash 13676 df-cj 14438 df-re 14439 df-im 14440 df-sqrt 14574 df-abs 14575 df-clim 14825 df-sum 15023 df-dvds 15588 df-bits 15749 |
This theorem is referenced by: bitsf1 15773 sadcaddlem 15784 sadcadd 15785 sadadd2lem 15786 sadadd2 15787 sadadd3 15788 sadaddlem 15793 sadasslem 15797 sadeq 15799 eulerpartgbij 31632 eulerpartlemmf 31635 |
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