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Theorem ackbij1b 8111
Description: The Ackermann bijection, part 1b: the bijection from ackbij1 8110 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 8091 . . . . 5  |-  ( A  e.  om  ->  ~P A  C_  ( ~P om  i^i  Fin ) )
2 pwexg 4375 . . . . 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 8108 . . . . . 6  |-  F :
( ~P om  i^i  Fin ) -1-1-> om
5 f1imaeng 7159 . . . . . 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 7291 . . . . . 6  |-  ( A  e.  om  ->  A  e.  Fin )
9 pwfi 7394 . . . . . 6  |-  ( A  e.  Fin  <->  ~P A  e.  Fin )
108, 9sylib 189 . . . . 5  |-  ( A  e.  om  ->  ~P A  e.  Fin )
11 ficardid 7841 . . . . 5  |-  ( ~P A  e.  Fin  ->  (
card `  ~P A ) 
~~  ~P A )
12 ensym 7148 . . . . 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 7151 . . . 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 7290 . . . . . . 7  |-  om  =  ( On  i^i  Fin )
17 inss2 3554 . . . . . . 7  |-  ( On 
i^i  Fin )  C_  Fin
1816, 17eqsstri 3370 . . . . . 6  |-  om  C_  Fin
19 ficardom 7840 . . . . . . 7  |-  ( ~P A  e.  Fin  ->  (
card `  ~P A )  e.  om )
2010, 19syl 16 . . . . . 6  |-  ( A  e.  om  ->  ( card `  ~P A )  e.  om )
2118, 20sseldi 3338 . . . . 5  |-  ( A  e.  om  ->  ( card `  ~P A )  e.  Fin )
22 php3 7285 . . . . . 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 7128 . . . 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 5734 . . . . . 6  |-  ( F `
 a )  e. 
_V
29 ackbij1lem3 8094 . . . . . . . . 9  |-  ( A  e.  om  ->  A  e.  ( ~P om  i^i  Fin ) )
30 elpwi 3799 . . . . . . . . 9  |-  ( a  e.  ~P A  -> 
a  C_  A )
313ackbij1lem12 8103 . . . . . . . . 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 8101 . . . . . . . . . . 11  |-  F :
( ~P om  i^i  Fin ) --> om
34 peano1 4856 . . . . . . . . . . 11  |-  (/)  e.  om
3533, 34f0cli 5872 . . . . . . . . . 10  |-  ( F `
 a )  e. 
om
36 nnord 4845 . . . . . . . . . 10  |-  ( ( F `  a )  e.  om  ->  Ord  ( F `  a ) )
3735, 36ax-mp 8 . . . . . . . . 9  |-  Ord  ( F `  a )
3833, 34f0cli 5872 . . . . . . . . . 10  |-  ( F `
 A )  e. 
om
39 nnord 4845 . . . . . . . . . 10  |-  ( ( F `  A )  e.  om  ->  Ord  ( F `  A ) )
4038, 39ax-mp 8 . . . . . . . . 9  |-  Ord  ( F `  A )
41 ordsucsssuc 4795 . . . . . . . . 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 8105 . . . . . . . . 9  |-  ( A  e.  om  ->  ( F `  { A } )  =  suc  ( F `  A ) )
453ackbij1lem8 8099 . . . . . . . . 9  |-  ( A  e.  om  ->  ( F `  { A } )  =  (
card `  ~P A ) )
4644, 45eqtr3d 2469 . . . . . . . 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 3376 . . . . . 6  |-  ( ( A  e.  om  /\  a  e.  ~P A
)  ->  suc  ( F `
 a )  C_  ( card `  ~P A ) )
49 sucssel 4666 . . . . . 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 2781 . . . 4  |-  ( A  e.  om  ->  A. a  e.  ~P  A ( F `
 a )  e.  ( card `  ~P A ) )
52 f1fun 5633 . . . . . 6  |-  ( F : ( ~P om  i^i  Fin ) -1-1-> om  ->  Fun 
F )
534, 52ax-mp 8 . . . . 5  |-  Fun  F
54 f1dm 5635 . . . . . . 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 3387 . . . . 5  |-  ( A  e.  om  ->  ~P A  C_  dom  F )
57 funimass4 5769 . . . . 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 3438 . . 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 1652    e. wcel 1725   A.wral 2697   _Vcvv 2948    i^i cin 3311    C_ wss 3312    C. wpss 3313   ~Pcpw 3791   {csn 3806   U_ciun 4085   class class class wbr 4204    e. cmpt 4258   Ord word 4572   Oncon0 4573   suc csuc 4575   omcom 4837    X. cxp 4868   dom cdm 4870   "cima 4873   Fun wfun 5440   -1-1->wf1 5443   ` cfv 5446    ~~ cen 7098    ~< csdm 7100   Fincfn 7101   cardccrd 7814
This theorem is referenced by:  ackbij2lem2  8112
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 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-card 7818  df-cda 8040
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