Theorem ackbij1 9060

Description: The Ackermann bijection, part 1: each natural number can be uniquely coded in binary as a finite set of natural numbers and conversely. (Contributed by Stefan O'Rear, 18-Nov-2014.)

Hypothesis

Ref	Expression
ackbij.f	|- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )

Assertion

Ref	Expression
ackbij1	|- F : ( ~P om i^i Fin ) -1-1-onto-> om

Distinct variable group:   x, F, y

Proof of Theorem ackbij1

Dummy variables a b c are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ackbij.f	. . 3 |- F = ( x e. ( ~P om i^i Fin ) |-> ( card ` U_ y e. x ( { y } X. ~P y ) ) )
2	1	ackbij1lem17 9058	. 2 |- F : ( ~P om i^i Fin ) -1-1-> om
3		f1f 6101	. . . 4 |- ( F : ( ~P om i^i Fin ) -1-1-> om -> F : ( ~P om i^i Fin ) --> om )
4		frn 6053	. . . 4 |- ( F : ( ~P om i^i Fin ) --> om -> ran F C_ om )
5	2, 3, 4	mp2b 10	. . 3 |- ran F C_ om
6		eleq1 2689	. . . . 5 |- ( b = (/) -> ( b e. ran F <-> (/) e. ran F ) )
7		eleq1 2689	. . . . 5 |- ( b = a -> ( b e. ran F <-> a e. ran F ) )
8		eleq1 2689	. . . . 5 |- ( b = suc a -> ( b e. ran F <-> suc a e. ran F ) )
9		peano1 7085	. . . . . . . 8 |- (/) e. om
10		ackbij1lem3 9044	. . . . . . . 8 |- ( (/) e. om -> (/) e. ( ~P om i^i Fin ) )
11	9, 10	ax-mp 5	. . . . . . 7 |- (/) e. ( ~P om i^i Fin )
12	1	ackbij1lem13 9054	. . . . . . 7 |- ( F ` (/) ) = (/)
13		fveq2 6191	. . . . . . . . 9 |- ( a = (/) -> ( F ` a ) = ( F ` (/) ) )
14	13	eqeq1d 2624	. . . . . . . 8 |- ( a = (/) -> ( ( F ` a ) = (/) <-> ( F ` (/) ) = (/) ) )
15	14	rspcev 3309	. . . . . . 7 |- ( ( (/) e. ( ~P om i^i Fin ) /\ ( F ` (/) ) = (/) ) -> E. a e. ( ~P om i^i Fin ) ( F ` a ) = (/) )
16	11, 12, 15	mp2an 708	. . . . . 6 |- E. a e. ( ~P om i^i Fin ) ( F ` a ) = (/)
17		f1fn 6102	. . . . . . . 8 |- ( F : ( ~P om i^i Fin ) -1-1-> om -> F Fn ( ~P om i^i Fin ) )
18	2, 17	ax-mp 5	. . . . . . 7 |- F Fn ( ~P om i^i Fin )
19		fvelrnb 6243	. . . . . . 7 |- ( F Fn ( ~P om i^i Fin ) -> ( (/) e. ran F <-> E. a e. ( ~P om i^i Fin ) ( F ` a ) = (/) ) )
20	18, 19	ax-mp 5	. . . . . 6 |- ( (/) e. ran F <-> E. a e. ( ~P om i^i Fin ) ( F ` a ) = (/) )
21	16, 20	mpbir 221	. . . . 5 |- (/) e. ran F
22	1	ackbij1lem18 9059	. . . . . . . . 9 |- ( c e. ( ~P om i^i Fin ) -> E. b e. ( ~P om i^i Fin ) ( F ` b ) = suc ( F ` c ) )
23	22	adantl 482	. . . . . . . 8 |- ( ( a e. om /\ c e. ( ~P om i^i Fin ) ) -> E. b e. ( ~P om i^i Fin ) ( F ` b ) = suc ( F ` c ) )
24		suceq 5790	. . . . . . . . . 10 |- ( ( F ` c ) = a -> suc ( F ` c ) = suc a )
25	24	eqeq2d 2632	. . . . . . . . 9 |- ( ( F ` c ) = a -> ( ( F ` b ) = suc ( F ` c ) <-> ( F ` b ) = suc a ) )
26	25	rexbidv 3052	. . . . . . . 8 |- ( ( F ` c ) = a -> ( E. b e. ( ~P om i^i Fin ) ( F ` b ) = suc ( F ` c ) <-> E. b e. ( ~P om i^i Fin ) ( F ` b ) = suc a ) )
27	23, 26	syl5ibcom 235	. . . . . . 7 |- ( ( a e. om /\ c e. ( ~P om i^i Fin ) ) -> ( ( F ` c ) = a -> E. b e. ( ~P om i^i Fin ) ( F ` b ) = suc a ) )
28	27	rexlimdva 3031	. . . . . 6 |- ( a e. om -> ( E. c e. ( ~P om i^i Fin ) ( F ` c ) = a -> E. b e. ( ~P om i^i Fin ) ( F ` b ) = suc a ) )
29		fvelrnb 6243	. . . . . . 7 |- ( F Fn ( ~P om i^i Fin ) -> ( a e. ran F <-> E. c e. ( ~P om i^i Fin ) ( F ` c ) = a ) )
30	18, 29	ax-mp 5	. . . . . 6 |- ( a e. ran F <-> E. c e. ( ~P om i^i Fin ) ( F ` c ) = a )
31		fvelrnb 6243	. . . . . . 7 |- ( F Fn ( ~P om i^i Fin ) -> ( suc a e. ran F <-> E. b e. ( ~P om i^i Fin ) ( F ` b ) = suc a ) )
32	18, 31	ax-mp 5	. . . . . 6 |- ( suc a e. ran F <-> E. b e. ( ~P om i^i Fin ) ( F ` b ) = suc a )
33	28, 30, 32	3imtr4g 285	. . . . 5 |- ( a e. om -> ( a e. ran F -> suc a e. ran F ) )
34	6, 7, 8, 7, 21, 33	finds 7092	. . . 4 |- ( a e. om -> a e. ran F )
35	34	ssriv 3607	. . 3 |- om C_ ran F
36	5, 35	eqssi 3619	. 2 |- ran F = om
37		dff1o5 6146	. 2 |- ( F : ( ~P om i^i Fin ) -1-1-onto-> om <-> ( F : ( ~P om i^i Fin ) -1-1-> om /\ ran F = om ) )
38	2, 36, 37	mpbir2an 955	1 |- F : ( ~P om i^i Fin ) -1-1-onto-> om

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   i^i cin 3573   C_ wss 3574   (/)c0 3915   ~Pcpw 4158   {csn 4177   U_ciun 4520   |-> cmpt 4729   X. cxp 5112   ran crn 5115   suc csuc 5725   Fn wfn 5883   -->wf 5884   -1-1->wf1 5885   -1-1-onto->wf1o 5887   ` cfv 5888   omcom 7065   Fincfn 7955   cardccrd 8761

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  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-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-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-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-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-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990

This theorem is referenced by:  fictb  9067  ackbijnn  14560