Theorem bitsf1ocnv 15166

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 14560. (Contributed by Mario Carneiro, 8-Sep-2016.)

Assertion

Ref	Expression
bitsf1ocnv	|- ( (bits |` NN0 ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin ) /\ `' (bits |` NN0 ) = ( x e. ( ~P NN0 i^i Fin ) |-> sum_ n e. x ( 2 ^ n ) ) )

Distinct variable group:   x, n

Proof of Theorem bitsf1ocnv

Dummy variable k is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . . . 6 |- ( k e. NN0 |-> (bits ` k ) ) = ( k e. NN0 |-> (bits ` k ) )
2		bitsss 15148	. . . . . . . . 9 |- (bits ` k ) C_ NN0
3	2	a1i 11	. . . . . . . 8 |- ( k e. NN0 -> (bits ` k ) C_ NN0 )
4		bitsfi 15159	. . . . . . . 8 |- ( k e. NN0 -> (bits ` k ) e. Fin )
5		elfpw 8268	. . . . . . . 8 |- ( (bits ` k ) e. ( ~P NN0 i^i Fin ) <-> ( (bits ` k ) C_ NN0 /\ (bits ` k ) e. Fin ) )
6	3, 4, 5	sylanbrc 698	. . . . . . 7 |- ( k e. NN0 -> (bits ` k ) e. ( ~P NN0 i^i Fin ) )
7	6	adantl 482	. . . . . 6 |- ( ( T. /\ k e. NN0 ) -> (bits ` k ) e. ( ~P NN0 i^i Fin ) )
8		elfpw 8268	. . . . . . . . 9 |- ( x e. ( ~P NN0 i^i Fin ) <-> ( x C_ NN0 /\ x e. Fin ) )
9	8	simprbi 480	. . . . . . . 8 |- ( x e. ( ~P NN0 i^i Fin ) -> x e. Fin )
10		2nn0 11309	. . . . . . . . . 10 |- 2 e. NN0
11	10	a1i 11	. . . . . . . . 9 |- ( ( x e. ( ~P NN0 i^i Fin ) /\ n e. x ) -> 2 e. NN0 )
12	8	simplbi 476	. . . . . . . . . 10 |- ( x e. ( ~P NN0 i^i Fin ) -> x C_ NN0 )
13	12	sselda 3603	. . . . . . . . 9 |- ( ( x e. ( ~P NN0 i^i Fin ) /\ n e. x ) -> n e. NN0 )
14	11, 13	nn0expcld 13031	. . . . . . . 8 |- ( ( x e. ( ~P NN0 i^i Fin ) /\ n e. x ) -> ( 2 ^ n ) e. NN0 )
15	9, 14	fsumnn0cl 14467	. . . . . . 7 |- ( x e. ( ~P NN0 i^i Fin ) -> sum_ n e. x ( 2 ^ n ) e. NN0 )
16	15	adantl 482	. . . . . 6 |- ( ( T. /\ x e. ( ~P NN0 i^i Fin ) ) -> sum_ n e. x ( 2 ^ n ) e. NN0 )
17		bitsinv2 15165	. . . . . . . . . 10 |- ( x e. ( ~P NN0 i^i Fin ) -> (bits ` sum_ n e. x ( 2 ^ n ) ) = x )
18	17	eqcomd 2628	. . . . . . . . 9 |- ( x e. ( ~P NN0 i^i Fin ) -> x = (bits ` sum_ n e. x ( 2 ^ n ) ) )
19	18	ad2antll 765	. . . . . . . 8 |- ( ( T. /\ ( k e. NN0 /\ x e. ( ~P NN0 i^i Fin ) ) ) -> x = (bits ` sum_ n e. x ( 2 ^ n ) ) )
20		fveq2 6191	. . . . . . . . 9 |- ( k = sum_ n e. x ( 2 ^ n ) -> (bits ` k ) = (bits ` sum_ n e. x ( 2 ^ n ) ) )
21	20	eqeq2d 2632	. . . . . . . 8 |- ( k = sum_ n e. x ( 2 ^ n ) -> ( x = (bits ` k ) <-> x = (bits ` sum_ n e. x ( 2 ^ n ) ) ) )
22	19, 21	syl5ibrcom 237	. . . . . . 7 |- ( ( T. /\ ( k e. NN0 /\ x e. ( ~P NN0 i^i Fin ) ) ) -> ( k = sum_ n e. x ( 2 ^ n ) -> x = (bits ` k ) ) )
23		bitsinv1 15164	. . . . . . . . . 10 |- ( k e. NN0 -> sum_ n e. (bits ` k ) ( 2 ^ n ) = k )
24	23	eqcomd 2628	. . . . . . . . 9 |- ( k e. NN0 -> k = sum_ n e. (bits ` k ) ( 2 ^ n ) )
25	24	ad2antrl 764	. . . . . . . 8 |- ( ( T. /\ ( k e. NN0 /\ x e. ( ~P NN0 i^i Fin ) ) ) -> k = sum_ n e. (bits ` k ) ( 2 ^ n ) )
26		sumeq1 14419	. . . . . . . . 9 |- ( x = (bits ` k ) -> sum_ n e. x ( 2 ^ n ) = sum_ n e. (bits ` k ) ( 2 ^ n ) )
27	26	eqeq2d 2632	. . . . . . . 8 |- ( x = (bits ` k ) -> ( k = sum_ n e. x ( 2 ^ n ) <-> k = sum_ n e. (bits ` k ) ( 2 ^ n ) ) )
28	25, 27	syl5ibrcom 237	. . . . . . 7 |- ( ( T. /\ ( k e. NN0 /\ x e. ( ~P NN0 i^i Fin ) ) ) -> ( x = (bits ` k ) -> k = sum_ n e. x ( 2 ^ n ) ) )
29	22, 28	impbid 202	. . . . . 6 |- ( ( T. /\ ( k e. NN0 /\ x e. ( ~P NN0 i^i Fin ) ) ) -> ( k = sum_ n e. x ( 2 ^ n ) <-> x = (bits ` k ) ) )
30	1, 7, 16, 29	f1ocnv2d 6886	. . . . 5 |- ( T. -> ( ( k e. NN0 |-> (bits ` k ) ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin ) /\ `' ( k e. NN0 |-> (bits ` k ) ) = ( x e. ( ~P NN0 i^i Fin ) |-> sum_ n e. x ( 2 ^ n ) ) ) )
31	30	simpld 475	. . . 4 |- ( T. -> ( k e. NN0 |-> (bits ` k ) ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin ) )
32		bitsf 15149	. . . . . . . . 9 |- bits : ZZ --> ~P NN0
33	32	a1i 11	. . . . . . . 8 |- ( T. -> bits : ZZ --> ~P NN0 )
34	33	feqmptd 6249	. . . . . . 7 |- ( T. -> bits = ( k e. ZZ |-> (bits ` k ) ) )
35	34	reseq1d 5395	. . . . . 6 |- ( T. -> (bits |` NN0 ) = ( ( k e. ZZ |-> (bits ` k ) ) |` NN0 ) )
36		nn0ssz 11398	. . . . . . 7 |- NN0 C_ ZZ
37		resmpt 5449	. . . . . . 7 |- ( NN0 C_ ZZ -> ( ( k e. ZZ |-> (bits ` k ) ) |` NN0 ) = ( k e. NN0 |-> (bits ` k ) ) )
38	36, 37	ax-mp 5	. . . . . 6 |- ( ( k e. ZZ |-> (bits ` k ) ) |` NN0 ) = ( k e. NN0 |-> (bits ` k ) )
39	35, 38	syl6eq 2672	. . . . 5 |- ( T. -> (bits |` NN0 ) = ( k e. NN0 |-> (bits ` k ) ) )
40		f1oeq1 6127	. . . . 5 |- ( (bits |` NN0 ) = ( k e. NN0 |-> (bits ` k ) ) -> ( (bits |` NN0 ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin ) <-> ( k e. NN0 |-> (bits ` k ) ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin ) ) )
41	39, 40	syl 17	. . . 4 |- ( T. -> ( (bits |` NN0 ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin ) <-> ( k e. NN0 |-> (bits ` k ) ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin ) ) )
42	31, 41	mpbird 247	. . 3 |- ( T. -> (bits |` NN0 ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin ) )
43	39	cnveqd 5298	. . . 4 |- ( T. -> `' (bits |` NN0 ) = `' ( k e. NN0 |-> (bits ` k ) ) )
44	30	simprd 479	. . . 4 |- ( T. -> `' ( k e. NN0 |-> (bits ` k ) ) = ( x e. ( ~P NN0 i^i Fin ) |-> sum_ n e. x ( 2 ^ n ) ) )
45	43, 44	eqtrd 2656	. . 3 |- ( T. -> `' (bits |` NN0 ) = ( x e. ( ~P NN0 i^i Fin ) |-> sum_ n e. x ( 2 ^ n ) ) )
46	42, 45	jca 554	. 2 |- ( T. -> ( (bits |` NN0 ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin ) /\ `' (bits |` NN0 ) = ( x e. ( ~P NN0 i^i Fin ) |-> sum_ n e. x ( 2 ^ n ) ) ) )
47	46	trud 1493	1 |- ( (bits |` NN0 ) : NN0 -1-1-onto-> ( ~P NN0 i^i Fin ) /\ `' (bits |` NN0 ) = ( x e. ( ~P NN0 i^i Fin ) |-> sum_ n e. x ( 2 ^ n ) ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   |-> cmpt 4729   `'ccnv 5113   |` cres 5116   -->wf 5884   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   Fincfn 7955   2c2 11070   NN0cn0 11292   ZZcz 11377   ^cexp 12860   sum_csu 14416  bitscbits 15141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-disj 4621  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-fl 12593  df-mod 12669  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-dvds 14984  df-bits 15144

This theorem is referenced by:  bitsf1o  15167  bitsinv  15170