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Theorem ackbij1b 8052
Description: The Ackermann bijection, part 1b: the bijection from ackbij1 8051 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 8032 . . . . 5  |-  ( A  e.  om  ->  ~P A  C_  ( ~P om  i^i  Fin ) )
2 pwexg 4324 . . . . 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 8049 . . . . . 6  |-  F :
( ~P om  i^i  Fin ) -1-1-> om
5 f1imaeng 7103 . . . . . 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 7235 . . . . . 6  |-  ( A  e.  om  ->  A  e.  Fin )
9 pwfi 7337 . . . . . 6  |-  ( A  e.  Fin  <->  ~P A  e.  Fin )
108, 9sylib 189 . . . . 5  |-  ( A  e.  om  ->  ~P A  e.  Fin )
11 ficardid 7782 . . . . 5  |-  ( ~P A  e.  Fin  ->  (
card `  ~P A ) 
~~  ~P A )
12 ensym 7092 . . . . 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 7095 . . . 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 7234 . . . . . . 7  |-  om  =  ( On  i^i  Fin )
17 inss2 3505 . . . . . . 7  |-  ( On 
i^i  Fin )  C_  Fin
1816, 17eqsstri 3321 . . . . . 6  |-  om  C_  Fin
19 ficardom 7781 . . . . . . 7  |-  ( ~P A  e.  Fin  ->  (
card `  ~P A )  e.  om )
2010, 19syl 16 . . . . . 6  |-  ( A  e.  om  ->  ( card `  ~P A )  e.  om )
2118, 20sseldi 3289 . . . . 5  |-  ( A  e.  om  ->  ( card `  ~P A )  e.  Fin )
22 php3 7229 . . . . . 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 7072 . . . 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 5682 . . . . . 6  |-  ( F `
 a )  e. 
_V
29 ackbij1lem3 8035 . . . . . . . . 9  |-  ( A  e.  om  ->  A  e.  ( ~P om  i^i  Fin ) )
30 elpwi 3750 . . . . . . . . 9  |-  ( a  e.  ~P A  -> 
a  C_  A )
313ackbij1lem12 8044 . . . . . . . . 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 8042 . . . . . . . . . . 11  |-  F :
( ~P om  i^i  Fin ) --> om
34 peano1 4804 . . . . . . . . . . 11  |-  (/)  e.  om
3533, 34f0cli 5819 . . . . . . . . . 10  |-  ( F `
 a )  e. 
om
36 nnord 4793 . . . . . . . . . 10  |-  ( ( F `  a )  e.  om  ->  Ord  ( F `  a ) )
3735, 36ax-mp 8 . . . . . . . . 9  |-  Ord  ( F `  a )
3833, 34f0cli 5819 . . . . . . . . . 10  |-  ( F `
 A )  e. 
om
39 nnord 4793 . . . . . . . . . 10  |-  ( ( F `  A )  e.  om  ->  Ord  ( F `  A ) )
4038, 39ax-mp 8 . . . . . . . . 9  |-  Ord  ( F `  A )
41 ordsucsssuc 4743 . . . . . . . . 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 8046 . . . . . . . . 9  |-  ( A  e.  om  ->  ( F `  { A } )  =  suc  ( F `  A ) )
453ackbij1lem8 8040 . . . . . . . . 9  |-  ( A  e.  om  ->  ( F `  { A } )  =  (
card `  ~P A ) )
4644, 45eqtr3d 2421 . . . . . . . 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 3327 . . . . . 6  |-  ( ( A  e.  om  /\  a  e.  ~P A
)  ->  suc  ( F `
 a )  C_  ( card `  ~P A ) )
49 sucssel 4614 . . . . . 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 2732 . . . 4  |-  ( A  e.  om  ->  A. a  e.  ~P  A ( F `
 a )  e.  ( card `  ~P A ) )
52 f1fun 5581 . . . . . 6  |-  ( F : ( ~P om  i^i  Fin ) -1-1-> om  ->  Fun 
F )
534, 52ax-mp 8 . . . . 5  |-  Fun  F
54 f1dm 5583 . . . . . . 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 3338 . . . . 5  |-  ( A  e.  om  ->  ~P A  C_  dom  F )
57 funimass4 5716 . . . . 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 3389 . . 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 1717   A.wral 2649   _Vcvv 2899    i^i cin 3262    C_ wss 3263    C. wpss 3264   ~Pcpw 3742   {csn 3757   U_ciun 4035   class class class wbr 4153    e. cmpt 4207   Ord word 4521   Oncon0 4522   suc csuc 4524   omcom 4785    X. cxp 4816   dom cdm 4818   "cima 4821   Fun wfun 5388   -1-1->wf1 5391   ` cfv 5394    ~~ cen 7042    ~< csdm 7044   Fincfn 7045   cardccrd 7755
This theorem is referenced by:  ackbij2lem2  8053
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rmo 2657  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-pss 3279  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-suc 4528  df-om 4786  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-ov 6023  df-oprab 6024  df-mpt2 6025  df-1st 6288  df-2nd 6289  df-recs 6569  df-rdg 6604  df-1o 6660  df-2o 6661  df-oadd 6664  df-er 6841  df-map 6956  df-en 7046  df-dom 7047  df-sdom 7048  df-fin 7049  df-card 7759  df-cda 7981
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