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Theorem ackbij1b 7865
Description: The Ackermann bijection, part 1b: the bijection from ackbij1 7864 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 7845 . . . . 5  |-  ( A  e.  om  ->  ~P A  C_  ( ~P om  i^i  Fin ) )
2 pwexg 4194 . . . . 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 7862 . . . . . 6  |-  F :
( ~P om  i^i  Fin ) -1-1-> om
5 f1imaeng 6921 . . . . . 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 1264 . . . . 5  |-  ( ( ~P A  C_  ( ~P om  i^i  Fin )  /\  ~P A  e.  _V )  ->  ( F " ~P A )  ~~  ~P A )
71, 2, 6syl2anc 642 . . . 4  |-  ( A  e.  om  ->  ( F " ~P A ) 
~~  ~P A )
8 nnfi 7053 . . . . . 6  |-  ( A  e.  om  ->  A  e.  Fin )
9 pwfi 7151 . . . . . 6  |-  ( A  e.  Fin  <->  ~P A  e.  Fin )
108, 9sylib 188 . . . . 5  |-  ( A  e.  om  ->  ~P A  e.  Fin )
11 ficardid 7595 . . . . 5  |-  ( ~P A  e.  Fin  ->  (
card `  ~P A ) 
~~  ~P A )
12 ensym 6910 . . . . 5  |-  ( (
card `  ~P A ) 
~~  ~P A  ->  ~P A  ~~  ( card `  ~P A ) )
1310, 11, 123syl 18 . . . 4  |-  ( A  e.  om  ->  ~P A  ~~  ( card `  ~P A ) )
14 entr 6913 . . . 4  |-  ( ( ( F " ~P A )  ~~  ~P A  /\  ~P A  ~~  ( card `  ~P A ) )  ->  ( F " ~P A )  ~~  ( card `  ~P A ) )
157, 13, 14syl2anc 642 . . 3  |-  ( A  e.  om  ->  ( F " ~P A ) 
~~  ( card `  ~P A ) )
16 onfin2 7052 . . . . . . 7  |-  om  =  ( On  i^i  Fin )
17 inss2 3390 . . . . . . 7  |-  ( On 
i^i  Fin )  C_  Fin
1816, 17eqsstri 3208 . . . . . 6  |-  om  C_  Fin
19 ficardom 7594 . . . . . . 7  |-  ( ~P A  e.  Fin  ->  (
card `  ~P A )  e.  om )
2010, 19syl 15 . . . . . 6  |-  ( A  e.  om  ->  ( card `  ~P A )  e.  om )
2118, 20sseldi 3178 . . . . 5  |-  ( A  e.  om  ->  ( card `  ~P A )  e.  Fin )
22 php3 7047 . . . . . 6  |-  ( ( ( card `  ~P A )  e.  Fin  /\  ( F " ~P A )  C.  ( card `  ~P A ) )  ->  ( F " ~P A )  ~< 
( card `  ~P A ) )
2322ex 423 . . . . 5  |-  ( (
card `  ~P A )  e.  Fin  ->  (
( F " ~P A )  C.  ( card `  ~P A )  ->  ( F " ~P A )  ~<  ( card `  ~P A ) ) )
2421, 23syl 15 . . . 4  |-  ( A  e.  om  ->  (
( F " ~P A )  C.  ( card `  ~P A )  ->  ( F " ~P A )  ~<  ( card `  ~P A ) ) )
25 sdomnen 6890 . . . 4  |-  ( ( F " ~P A
)  ~<  ( card `  ~P A )  ->  -.  ( F " ~P A
)  ~~  ( card `  ~P A ) )
2624, 25syl6 29 . . 3  |-  ( A  e.  om  ->  (
( F " ~P A )  C.  ( card `  ~P A )  ->  -.  ( F " ~P A )  ~~  ( card `  ~P A ) ) )
2715, 26mt2d 109 . 2  |-  ( A  e.  om  ->  -.  ( F " ~P A
)  C.  ( card `  ~P A ) )
28 fvex 5539 . . . . . 6  |-  ( F `
 a )  e. 
_V
29 ackbij1lem3 7848 . . . . . . . . 9  |-  ( A  e.  om  ->  A  e.  ( ~P om  i^i  Fin ) )
30 elpwi 3633 . . . . . . . . 9  |-  ( a  e.  ~P A  -> 
a  C_  A )
313ackbij1lem12 7857 . . . . . . . . 9  |-  ( ( A  e.  ( ~P
om  i^i  Fin )  /\  a  C_  A )  ->  ( F `  a )  C_  ( F `  A )
)
3229, 30, 31syl2an 463 . . . . . . . 8  |-  ( ( A  e.  om  /\  a  e.  ~P A
)  ->  ( F `  a )  C_  ( F `  A )
)
333ackbij1lem10 7855 . . . . . . . . . . 11  |-  F :
( ~P om  i^i  Fin ) --> om
34 peano1 4675 . . . . . . . . . . 11  |-  (/)  e.  om
3533, 34f0cli 5671 . . . . . . . . . 10  |-  ( F `
 a )  e. 
om
36 nnord 4664 . . . . . . . . . 10  |-  ( ( F `  a )  e.  om  ->  Ord  ( F `  a ) )
3735, 36ax-mp 8 . . . . . . . . 9  |-  Ord  ( F `  a )
3833, 34f0cli 5671 . . . . . . . . . 10  |-  ( F `
 A )  e. 
om
39 nnord 4664 . . . . . . . . . 10  |-  ( ( F `  A )  e.  om  ->  Ord  ( F `  A ) )
4038, 39ax-mp 8 . . . . . . . . 9  |-  Ord  ( F `  A )
41 ordsucsssuc 4614 . . . . . . . . 9  |-  ( ( Ord  ( F `  a )  /\  Ord  ( F `  A ) )  ->  ( ( F `  a )  C_  ( F `  A
)  <->  suc  ( F `  a )  C_  suc  ( F `  A ) ) )
4237, 40, 41mp2an 653 . . . . . . . 8  |-  ( ( F `  a ) 
C_  ( F `  A )  <->  suc  ( F `
 a )  C_  suc  ( F `  A
) )
4332, 42sylib 188 . . . . . . 7  |-  ( ( A  e.  om  /\  a  e.  ~P A
)  ->  suc  ( F `
 a )  C_  suc  ( F `  A
) )
443ackbij1lem14 7859 . . . . . . . . 9  |-  ( A  e.  om  ->  ( F `  { A } )  =  suc  ( F `  A ) )
453ackbij1lem8 7853 . . . . . . . . 9  |-  ( A  e.  om  ->  ( F `  { A } )  =  (
card `  ~P A ) )
4644, 45eqtr3d 2317 . . . . . . . 8  |-  ( A  e.  om  ->  suc  ( F `  A )  =  ( card `  ~P A ) )
4746adantr 451 . . . . . . 7  |-  ( ( A  e.  om  /\  a  e.  ~P A
)  ->  suc  ( F `
 A )  =  ( card `  ~P A ) )
4843, 47sseqtrd 3214 . . . . . 6  |-  ( ( A  e.  om  /\  a  e.  ~P A
)  ->  suc  ( F `
 a )  C_  ( card `  ~P A ) )
49 sucssel 4485 . . . . . 6  |-  ( ( F `  a )  e.  _V  ->  ( suc  ( F `  a
)  C_  ( card `  ~P A )  -> 
( F `  a
)  e.  ( card `  ~P A ) ) )
5028, 48, 49mpsyl 59 . . . . 5  |-  ( ( A  e.  om  /\  a  e.  ~P A
)  ->  ( F `  a )  e.  (
card `  ~P A ) )
5150ralrimiva 2626 . . . 4  |-  ( A  e.  om  ->  A. a  e.  ~P  A ( F `
 a )  e.  ( card `  ~P A ) )
52 f1fun 5439 . . . . . 6  |-  ( F : ( ~P om  i^i  Fin ) -1-1-> om  ->  Fun 
F )
534, 52ax-mp 8 . . . . 5  |-  Fun  F
54 f1dm 5441 . . . . . . 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 3225 . . . . 5  |-  ( A  e.  om  ->  ~P A  C_  dom  F )
57 funimass4 5573 . . . . 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 644 . . . 4  |-  ( A  e.  om  ->  (
( F " ~P A )  C_  ( card `  ~P A )  <->  A. a  e.  ~P  A ( F `  a )  e.  (
card `  ~P A ) ) )
5951, 58mpbird 223 . . 3  |-  ( A  e.  om  ->  ( F " ~P A ) 
C_  ( card `  ~P A ) )
60 sspss 3275 . . 3  |-  ( ( F " ~P A
)  C_  ( card `  ~P A )  <->  ( ( F " ~P A ) 
C.  ( card `  ~P A )  \/  ( F " ~P A )  =  ( card `  ~P A ) ) )
6159, 60sylib 188 . 2  |-  ( A  e.  om  ->  (
( F " ~P A )  C.  ( card `  ~P A )  \/  ( F " ~P A )  =  (
card `  ~P A ) ) )
62 orel1 371 . 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 56 1  |-  ( A  e.  om  ->  ( F " ~P A )  =  ( card `  ~P A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    i^i cin 3151    C_ wss 3152    C. wpss 3153   ~Pcpw 3625   {csn 3640   U_ciun 3905   class class class wbr 4023    e. cmpt 4077   Ord word 4391   Oncon0 4392   suc csuc 4394   omcom 4656    X. cxp 4687   dom cdm 4689   "cima 4692   Fun wfun 5249   -1-1->wf1 5252   ` cfv 5255    ~~ cen 6860    ~< csdm 6862   Fincfn 6863   cardccrd 7568
This theorem is referenced by:  ackbij2lem2  7866
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-card 7572  df-cda 7794
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