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Theorem ackbij1 8074
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 8072 . 2  |-  F :
( ~P om  i^i  Fin ) -1-1-> om
3 f1f 5598 . . . 4  |-  ( F : ( ~P om  i^i  Fin ) -1-1-> om  ->  F : ( ~P om  i^i  Fin ) --> om )
4 frn 5556 . . . 4  |-  ( F : ( ~P om  i^i  Fin ) --> om  ->  ran 
F  C_  om )
52, 3, 4mp2b 10 . . 3  |-  ran  F  C_ 
om
6 eleq1 2464 . . . . 5  |-  ( b  =  (/)  ->  ( b  e.  ran  F  <->  (/)  e.  ran  F ) )
7 eleq1 2464 . . . . 5  |-  ( b  =  a  ->  (
b  e.  ran  F  <->  a  e.  ran  F ) )
8 eleq1 2464 . . . . 5  |-  ( b  =  suc  a  -> 
( b  e.  ran  F  <->  suc  a  e.  ran  F ) )
9 peano1 4823 . . . . . . . 8  |-  (/)  e.  om
10 ackbij1lem3 8058 . . . . . . . 8  |-  ( (/)  e.  om  ->  (/)  e.  ( ~P om  i^i  Fin ) )
119, 10ax-mp 8 . . . . . . 7  |-  (/)  e.  ( ~P om  i^i  Fin )
121ackbij1lem13 8068 . . . . . . 7  |-  ( F `
 (/) )  =  (/)
13 fveq2 5687 . . . . . . . . 9  |-  ( a  =  (/)  ->  ( F `
 a )  =  ( F `  (/) ) )
1413eqeq1d 2412 . . . . . . . 8  |-  ( a  =  (/)  ->  ( ( F `  a )  =  (/)  <->  ( F `  (/) )  =  (/) ) )
1514rspcev 3012 . . . . . . 7  |-  ( (
(/)  e.  ( ~P om  i^i  Fin )  /\  ( F `  (/) )  =  (/) )  ->  E. a  e.  ( ~P om  i^i  Fin ) ( F `  a )  =  (/) )
1611, 12, 15mp2an 654 . . . . . 6  |-  E. a  e.  ( ~P om  i^i  Fin ) ( F `  a )  =  (/)
17 f1fn 5599 . . . . . . . 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 5733 . . . . . . 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 201 . . . . 5  |-  (/)  e.  ran  F
221ackbij1lem18 8073 . . . . . . . . 9  |-  ( c  e.  ( ~P om  i^i  Fin )  ->  E. b  e.  ( ~P om  i^i  Fin ) ( F `  b )  =  suc  ( F `  c ) )
2322adantl 453 . . . . . . . 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 4606 . . . . . . . . . 10  |-  ( ( F `  c )  =  a  ->  suc  ( F `  c )  =  suc  a )
2524eqeq2d 2415 . . . . . . . . 9  |-  ( ( F `  c )  =  a  ->  (
( F `  b
)  =  suc  ( F `  c )  <->  ( F `  b )  =  suc  a ) )
2625rexbidv 2687 . . . . . . . 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 212 . . . . . . 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 2790 . . . . . 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 5733 . . . . . . 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 5733 . . . . . . 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 262 . . . . 5  |-  ( a  e.  om  ->  (
a  e.  ran  F  ->  suc  a  e.  ran  F ) )
346, 7, 8, 7, 21, 33finds 4830 . . . 4  |-  ( a  e.  om  ->  a  e.  ran  F )
3534ssriv 3312 . . 3  |-  om  C_  ran  F
365, 35eqssi 3324 . 2  |-  ran  F  =  om
37 dff1o5 5642 . 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 887 1  |-  F :
( ~P om  i^i  Fin ) -1-1-onto-> om
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   {csn 3774   U_ciun 4053    e. cmpt 4226   suc csuc 4543   omcom 4804    X. cxp 4835   ran crn 4838    Fn wfn 5408   -->wf 5409   -1-1->wf1 5410   -1-1-onto->wf1o 5412   ` cfv 5413   Fincfn 7068   cardccrd 7778
This theorem is referenced by:  fictb  8081  ackbijnn  12562
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-card 7782  df-cda 8004
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