MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ackbij1 Structured version   Unicode version

Theorem ackbij1 8118
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 8116 . 2  |-  F :
( ~P om  i^i  Fin ) -1-1-> om
3 f1f 5639 . . . 4  |-  ( F : ( ~P om  i^i  Fin ) -1-1-> om  ->  F : ( ~P om  i^i  Fin ) --> om )
4 frn 5597 . . . 4  |-  ( F : ( ~P om  i^i  Fin ) --> om  ->  ran 
F  C_  om )
52, 3, 4mp2b 10 . . 3  |-  ran  F  C_ 
om
6 eleq1 2496 . . . . 5  |-  ( b  =  (/)  ->  ( b  e.  ran  F  <->  (/)  e.  ran  F ) )
7 eleq1 2496 . . . . 5  |-  ( b  =  a  ->  (
b  e.  ran  F  <->  a  e.  ran  F ) )
8 eleq1 2496 . . . . 5  |-  ( b  =  suc  a  -> 
( b  e.  ran  F  <->  suc  a  e.  ran  F ) )
9 peano1 4864 . . . . . . . 8  |-  (/)  e.  om
10 ackbij1lem3 8102 . . . . . . . 8  |-  ( (/)  e.  om  ->  (/)  e.  ( ~P om  i^i  Fin ) )
119, 10ax-mp 8 . . . . . . 7  |-  (/)  e.  ( ~P om  i^i  Fin )
121ackbij1lem13 8112 . . . . . . 7  |-  ( F `
 (/) )  =  (/)
13 fveq2 5728 . . . . . . . . 9  |-  ( a  =  (/)  ->  ( F `
 a )  =  ( F `  (/) ) )
1413eqeq1d 2444 . . . . . . . 8  |-  ( a  =  (/)  ->  ( ( F `  a )  =  (/)  <->  ( F `  (/) )  =  (/) ) )
1514rspcev 3052 . . . . . . 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 5640 . . . . . . . 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 5774 . . . . . . 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 8117 . . . . . . . . 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 4646 . . . . . . . . . 10  |-  ( ( F `  c )  =  a  ->  suc  ( F `  c )  =  suc  a )
2524eqeq2d 2447 . . . . . . . . 9  |-  ( ( F `  c )  =  a  ->  (
( F `  b
)  =  suc  ( F `  c )  <->  ( F `  b )  =  suc  a ) )
2625rexbidv 2726 . . . . . . . 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 2830 . . . . . 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 5774 . . . . . . 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 5774 . . . . . . 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 4871 . . . 4  |-  ( a  e.  om  ->  a  e.  ran  F )
3534ssriv 3352 . . 3  |-  om  C_  ran  F
365, 35eqssi 3364 . 2  |-  ran  F  =  om
37 dff1o5 5683 . 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 1652    e. wcel 1725   E.wrex 2706    i^i cin 3319    C_ wss 3320   (/)c0 3628   ~Pcpw 3799   {csn 3814   U_ciun 4093    e. cmpt 4266   suc csuc 4583   omcom 4845    X. cxp 4876   ran crn 4879    Fn wfn 5449   -->wf 5450   -1-1->wf1 5451   -1-1-onto->wf1o 5453   ` cfv 5454   Fincfn 7109   cardccrd 7822
This theorem is referenced by:  fictb  8125  ackbijnn  12607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-cda 8048
  Copyright terms: Public domain W3C validator