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Theorem ackbij1 7909
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.
StepHypRef Expression
1 ackbij.f . . 3  |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ y  e.  x  ( {
y }  X.  ~P y ) ) )
21ackbij1lem17 7907 . 2  |-  F :
( ~P om  i^i  Fin ) -1-1-> om
3 f1f 5475 . . . 4  |-  ( F : ( ~P om  i^i  Fin ) -1-1-> om  ->  F : ( ~P om  i^i  Fin ) --> om )
4 frn 5433 . . . 4  |-  ( F : ( ~P om  i^i  Fin ) --> om  ->  ran 
F  C_  om )
52, 3, 4mp2b 9 . . 3  |-  ran  F  C_ 
om
6 eleq1 2376 . . . . 5  |-  ( b  =  (/)  ->  ( b  e.  ran  F  <->  (/)  e.  ran  F ) )
7 eleq1 2376 . . . . 5  |-  ( b  =  a  ->  (
b  e.  ran  F  <->  a  e.  ran  F ) )
8 eleq1 2376 . . . . 5  |-  ( b  =  suc  a  -> 
( b  e.  ran  F  <->  suc  a  e.  ran  F ) )
9 peano1 4712 . . . . . . . 8  |-  (/)  e.  om
10 ackbij1lem3 7893 . . . . . . . 8  |-  ( (/)  e.  om  ->  (/)  e.  ( ~P om  i^i  Fin ) )
119, 10ax-mp 8 . . . . . . 7  |-  (/)  e.  ( ~P om  i^i  Fin )
121ackbij1lem13 7903 . . . . . . 7  |-  ( F `
 (/) )  =  (/)
13 fveq2 5563 . . . . . . . . 9  |-  ( a  =  (/)  ->  ( F `
 a )  =  ( F `  (/) ) )
1413eqeq1d 2324 . . . . . . . 8  |-  ( a  =  (/)  ->  ( ( F `  a )  =  (/)  <->  ( F `  (/) )  =  (/) ) )
1514rspcev 2918 . . . . . . 7  |-  ( (
(/)  e.  ( ~P om  i^i  Fin )  /\  ( F `  (/) )  =  (/) )  ->  E. a  e.  ( ~P om  i^i  Fin ) ( F `  a )  =  (/) )
1611, 12, 15mp2an 653 . . . . . 6  |-  E. a  e.  ( ~P om  i^i  Fin ) ( F `  a )  =  (/)
17 f1fn 5476 . . . . . . . 8  |-  ( F : ( ~P om  i^i  Fin ) -1-1-> om  ->  F  Fn  ( ~P om  i^i  Fin ) )
182, 17ax-mp 8 . . . . . . 7  |-  F  Fn  ( ~P om  i^i  Fin )
19 fvelrnb 5608 . . . . . . 7  |-  ( F  Fn  ( ~P om  i^i  Fin )  ->  ( (/) 
e.  ran  F  <->  E. a  e.  ( ~P om  i^i  Fin ) ( F `  a )  =  (/) ) )
2018, 19ax-mp 8 . . . . . 6  |-  ( (/)  e.  ran  F  <->  E. a  e.  ( ~P om  i^i  Fin ) ( F `  a )  =  (/) )
2116, 20mpbir 200 . . . . 5  |-  (/)  e.  ran  F
221ackbij1lem18 7908 . . . . . . . . 9  |-  ( c  e.  ( ~P om  i^i  Fin )  ->  E. b  e.  ( ~P om  i^i  Fin ) ( F `  b )  =  suc  ( F `  c ) )
2322adantl 452 . . . . . . . 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 4494 . . . . . . . . . 10  |-  ( ( F `  c )  =  a  ->  suc  ( F `  c )  =  suc  a )
2524eqeq2d 2327 . . . . . . . . 9  |-  ( ( F `  c )  =  a  ->  (
( F `  b
)  =  suc  ( F `  c )  <->  ( F `  b )  =  suc  a ) )
2625rexbidv 2598 . . . . . . . 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
) )
2723, 26syl5ibcom 211 . . . . . . 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 ) )
2827rexlimdva 2701 . . . . . 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 5608 . . . . . . 7  |-  ( F  Fn  ( ~P om  i^i  Fin )  ->  (
a  e.  ran  F  <->  E. c  e.  ( ~P
om  i^i  Fin )
( F `  c
)  =  a ) )
3018, 29ax-mp 8 . . . . . 6  |-  ( a  e.  ran  F  <->  E. c  e.  ( ~P om  i^i  Fin ) ( F `  c )  =  a )
31 fvelrnb 5608 . . . . . . 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
) )
3218, 31ax-mp 8 . . . . . 6  |-  ( suc  a  e.  ran  F  <->  E. b  e.  ( ~P
om  i^i  Fin )
( F `  b
)  =  suc  a
)
3328, 30, 323imtr4g 261 . . . . 5  |-  ( a  e.  om  ->  (
a  e.  ran  F  ->  suc  a  e.  ran  F ) )
346, 7, 8, 7, 21, 33finds 4719 . . . 4  |-  ( a  e.  om  ->  a  e.  ran  F )
3534ssriv 3218 . . 3  |-  om  C_  ran  F
365, 35eqssi 3229 . 2  |-  ran  F  =  om
37 dff1o5 5519 . 2  |-  ( F : ( ~P om  i^i  Fin ) -1-1-onto-> om  <->  ( F :
( ~P om  i^i  Fin ) -1-1-> om  /\  ran  F  =  om ) )
382, 36, 37mpbir2an 886 1  |-  F :
( ~P om  i^i  Fin ) -1-1-onto-> om
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1633    e. wcel 1701   E.wrex 2578    i^i cin 3185    C_ wss 3186   (/)c0 3489   ~Pcpw 3659   {csn 3674   U_ciun 3942    e. cmpt 4114   suc csuc 4431   omcom 4693    X. cxp 4724   ran crn 4727    Fn wfn 5287   -->wf 5288   -1-1->wf1 5289   -1-1-onto->wf1o 5291   ` cfv 5292   Fincfn 6906   cardccrd 7613
This theorem is referenced by:  fictb  7916  ackbijnn  12333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-1st 6164  df-2nd 6165  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-er 6702  df-map 6817  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-card 7617  df-cda 7839
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