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Theorem ackbij1b 8075
Description: The Ackermann bijection, part 1b: the bijection from ackbij1 8074 restricts naturally to the powers of particular naturals. (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
ackbij1b  |-  ( A  e.  om  ->  ( F " ~P A )  =  ( card `  ~P A ) )
Distinct variable groups:    x, F, y    x, A, y

Proof of Theorem ackbij1b
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 ackbij2lem1 8055 . . . . 5  |-  ( A  e.  om  ->  ~P A  C_  ( ~P om  i^i  Fin ) )
2 pwexg 4343 . . . . 5  |-  ( A  e.  om  ->  ~P A  e.  _V )
3 ackbij.f . . . . . . 7  |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ y  e.  x  ( {
y }  X.  ~P y ) ) )
43ackbij1lem17 8072 . . . . . 6  |-  F :
( ~P om  i^i  Fin ) -1-1-> om
5 f1imaeng 7126 . . . . . 6  |-  ( ( F : ( ~P
om  i^i  Fin ) -1-1-> om  /\  ~P A  C_  ( ~P om  i^i  Fin )  /\  ~P A  e. 
_V )  ->  ( F " ~P A ) 
~~  ~P A )
64, 5mp3an1 1266 . . . . 5  |-  ( ( ~P A  C_  ( ~P om  i^i  Fin )  /\  ~P A  e.  _V )  ->  ( F " ~P A )  ~~  ~P A )
71, 2, 6syl2anc 643 . . . 4  |-  ( A  e.  om  ->  ( F " ~P A ) 
~~  ~P A )
8 nnfi 7258 . . . . . 6  |-  ( A  e.  om  ->  A  e.  Fin )
9 pwfi 7360 . . . . . 6  |-  ( A  e.  Fin  <->  ~P A  e.  Fin )
108, 9sylib 189 . . . . 5  |-  ( A  e.  om  ->  ~P A  e.  Fin )
11 ficardid 7805 . . . . 5  |-  ( ~P A  e.  Fin  ->  (
card `  ~P A ) 
~~  ~P A )
12 ensym 7115 . . . . 5  |-  ( (
card `  ~P A ) 
~~  ~P A  ->  ~P A  ~~  ( card `  ~P A ) )
1310, 11, 123syl 19 . . . 4  |-  ( A  e.  om  ->  ~P A  ~~  ( card `  ~P A ) )
14 entr 7118 . . . 4  |-  ( ( ( F " ~P A )  ~~  ~P A  /\  ~P A  ~~  ( card `  ~P A ) )  ->  ( F " ~P A )  ~~  ( card `  ~P A ) )
157, 13, 14syl2anc 643 . . 3  |-  ( A  e.  om  ->  ( F " ~P A ) 
~~  ( card `  ~P A ) )
16 onfin2 7257 . . . . . . 7  |-  om  =  ( On  i^i  Fin )
17 inss2 3522 . . . . . . 7  |-  ( On 
i^i  Fin )  C_  Fin
1816, 17eqsstri 3338 . . . . . 6  |-  om  C_  Fin
19 ficardom 7804 . . . . . . 7  |-  ( ~P A  e.  Fin  ->  (
card `  ~P A )  e.  om )
2010, 19syl 16 . . . . . 6  |-  ( A  e.  om  ->  ( card `  ~P A )  e.  om )
2118, 20sseldi 3306 . . . . 5  |-  ( A  e.  om  ->  ( card `  ~P A )  e.  Fin )
22 php3 7252 . . . . . 6  |-  ( ( ( card `  ~P A )  e.  Fin  /\  ( F " ~P A )  C.  ( card `  ~P A ) )  ->  ( F " ~P A )  ~< 
( card `  ~P A ) )
2322ex 424 . . . . 5  |-  ( (
card `  ~P A )  e.  Fin  ->  (
( F " ~P A )  C.  ( card `  ~P A )  ->  ( F " ~P A )  ~<  ( card `  ~P A ) ) )
2421, 23syl 16 . . . 4  |-  ( A  e.  om  ->  (
( F " ~P A )  C.  ( card `  ~P A )  ->  ( F " ~P A )  ~<  ( card `  ~P A ) ) )
25 sdomnen 7095 . . . 4  |-  ( ( F " ~P A
)  ~<  ( card `  ~P A )  ->  -.  ( F " ~P A
)  ~~  ( card `  ~P A ) )
2624, 25syl6 31 . . 3  |-  ( A  e.  om  ->  (
( F " ~P A )  C.  ( card `  ~P A )  ->  -.  ( F " ~P A )  ~~  ( card `  ~P A ) ) )
2715, 26mt2d 111 . 2  |-  ( A  e.  om  ->  -.  ( F " ~P A
)  C.  ( card `  ~P A ) )
28 fvex 5701 . . . . . 6  |-  ( F `
 a )  e. 
_V
29 ackbij1lem3 8058 . . . . . . . . 9  |-  ( A  e.  om  ->  A  e.  ( ~P om  i^i  Fin ) )
30 elpwi 3767 . . . . . . . . 9  |-  ( a  e.  ~P A  -> 
a  C_  A )
313ackbij1lem12 8067 . . . . . . . . 9  |-  ( ( A  e.  ( ~P
om  i^i  Fin )  /\  a  C_  A )  ->  ( F `  a )  C_  ( F `  A )
)
3229, 30, 31syl2an 464 . . . . . . . 8  |-  ( ( A  e.  om  /\  a  e.  ~P A
)  ->  ( F `  a )  C_  ( F `  A )
)
333ackbij1lem10 8065 . . . . . . . . . . 11  |-  F :
( ~P om  i^i  Fin ) --> om
34 peano1 4823 . . . . . . . . . . 11  |-  (/)  e.  om
3533, 34f0cli 5839 . . . . . . . . . 10  |-  ( F `
 a )  e. 
om
36 nnord 4812 . . . . . . . . . 10  |-  ( ( F `  a )  e.  om  ->  Ord  ( F `  a ) )
3735, 36ax-mp 8 . . . . . . . . 9  |-  Ord  ( F `  a )
3833, 34f0cli 5839 . . . . . . . . . 10  |-  ( F `
 A )  e. 
om
39 nnord 4812 . . . . . . . . . 10  |-  ( ( F `  A )  e.  om  ->  Ord  ( F `  A ) )
4038, 39ax-mp 8 . . . . . . . . 9  |-  Ord  ( F `  A )
41 ordsucsssuc 4762 . . . . . . . . 9  |-  ( ( Ord  ( F `  a )  /\  Ord  ( F `  A ) )  ->  ( ( F `  a )  C_  ( F `  A
)  <->  suc  ( F `  a )  C_  suc  ( F `  A ) ) )
4237, 40, 41mp2an 654 . . . . . . . 8  |-  ( ( F `  a ) 
C_  ( F `  A )  <->  suc  ( F `
 a )  C_  suc  ( F `  A
) )
4332, 42sylib 189 . . . . . . 7  |-  ( ( A  e.  om  /\  a  e.  ~P A
)  ->  suc  ( F `
 a )  C_  suc  ( F `  A
) )
443ackbij1lem14 8069 . . . . . . . . 9  |-  ( A  e.  om  ->  ( F `  { A } )  =  suc  ( F `  A ) )
453ackbij1lem8 8063 . . . . . . . . 9  |-  ( A  e.  om  ->  ( F `  { A } )  =  (
card `  ~P A ) )
4644, 45eqtr3d 2438 . . . . . . . 8  |-  ( A  e.  om  ->  suc  ( F `  A )  =  ( card `  ~P A ) )
4746adantr 452 . . . . . . 7  |-  ( ( A  e.  om  /\  a  e.  ~P A
)  ->  suc  ( F `
 A )  =  ( card `  ~P A ) )
4843, 47sseqtrd 3344 . . . . . 6  |-  ( ( A  e.  om  /\  a  e.  ~P A
)  ->  suc  ( F `
 a )  C_  ( card `  ~P A ) )
49 sucssel 4633 . . . . . 6  |-  ( ( F `  a )  e.  _V  ->  ( suc  ( F `  a
)  C_  ( card `  ~P A )  -> 
( F `  a
)  e.  ( card `  ~P A ) ) )
5028, 48, 49mpsyl 61 . . . . 5  |-  ( ( A  e.  om  /\  a  e.  ~P A
)  ->  ( F `  a )  e.  (
card `  ~P A ) )
5150ralrimiva 2749 . . . 4  |-  ( A  e.  om  ->  A. a  e.  ~P  A ( F `
 a )  e.  ( card `  ~P A ) )
52 f1fun 5600 . . . . . 6  |-  ( F : ( ~P om  i^i  Fin ) -1-1-> om  ->  Fun 
F )
534, 52ax-mp 8 . . . . 5  |-  Fun  F
54 f1dm 5602 . . . . . . 7  |-  ( F : ( ~P om  i^i  Fin ) -1-1-> om  ->  dom 
F  =  ( ~P
om  i^i  Fin )
)
554, 54ax-mp 8 . . . . . 6  |-  dom  F  =  ( ~P om  i^i  Fin )
561, 55syl6sseqr 3355 . . . . 5  |-  ( A  e.  om  ->  ~P A  C_  dom  F )
57 funimass4 5736 . . . . 5  |-  ( ( Fun  F  /\  ~P A  C_  dom  F )  ->  ( ( F
" ~P A ) 
C_  ( card `  ~P A )  <->  A. a  e.  ~P  A ( F `
 a )  e.  ( card `  ~P A ) ) )
5853, 56, 57sylancr 645 . . . 4  |-  ( A  e.  om  ->  (
( F " ~P A )  C_  ( card `  ~P A )  <->  A. a  e.  ~P  A ( F `  a )  e.  (
card `  ~P A ) ) )
5951, 58mpbird 224 . . 3  |-  ( A  e.  om  ->  ( F " ~P A ) 
C_  ( card `  ~P A ) )
60 sspss 3406 . . 3  |-  ( ( F " ~P A
)  C_  ( card `  ~P A )  <->  ( ( F " ~P A ) 
C.  ( card `  ~P A )  \/  ( F " ~P A )  =  ( card `  ~P A ) ) )
6159, 60sylib 189 . 2  |-  ( A  e.  om  ->  (
( F " ~P A )  C.  ( card `  ~P A )  \/  ( F " ~P A )  =  (
card `  ~P A ) ) )
62 orel1 372 . 2  |-  ( -.  ( F " ~P A )  C.  ( card `  ~P A )  ->  ( ( ( F " ~P A
)  C.  ( card `  ~P A )  \/  ( F " ~P A )  =  (
card `  ~P A ) )  ->  ( F " ~P A )  =  ( card `  ~P A ) ) )
6327, 61, 62sylc 58 1  |-  ( A  e.  om  ->  ( F " ~P A )  =  ( card `  ~P A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    i^i cin 3279    C_ wss 3280    C. wpss 3281   ~Pcpw 3759   {csn 3774   U_ciun 4053   class class class wbr 4172    e. cmpt 4226   Ord word 4540   Oncon0 4541   suc csuc 4543   omcom 4804    X. cxp 4835   dom cdm 4837   "cima 4840   Fun wfun 5407   -1-1->wf1 5410   ` cfv 5413    ~~ cen 7065    ~< csdm 7067   Fincfn 7068   cardccrd 7778
This theorem is referenced by:  ackbij2lem2  8076
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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