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Theorem ackbij1b 7881
Description: The Ackermann bijection, part 1b: the bijection from ackbij1 7880 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 7861 . . . . 5  |-  ( A  e.  om  ->  ~P A  C_  ( ~P om  i^i  Fin ) )
2 pwexg 4210 . . . . 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 7878 . . . . . 6  |-  F :
( ~P om  i^i  Fin ) -1-1-> om
5 f1imaeng 6937 . . . . . 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 7069 . . . . . 6  |-  ( A  e.  om  ->  A  e.  Fin )
9 pwfi 7167 . . . . . 6  |-  ( A  e.  Fin  <->  ~P A  e.  Fin )
108, 9sylib 188 . . . . 5  |-  ( A  e.  om  ->  ~P A  e.  Fin )
11 ficardid 7611 . . . . 5  |-  ( ~P A  e.  Fin  ->  (
card `  ~P A ) 
~~  ~P A )
12 ensym 6926 . . . . 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 6929 . . . 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 7068 . . . . . . 7  |-  om  =  ( On  i^i  Fin )
17 inss2 3403 . . . . . . 7  |-  ( On 
i^i  Fin )  C_  Fin
1816, 17eqsstri 3221 . . . . . 6  |-  om  C_  Fin
19 ficardom 7610 . . . . . . 7  |-  ( ~P A  e.  Fin  ->  (
card `  ~P A )  e.  om )
2010, 19syl 15 . . . . . 6  |-  ( A  e.  om  ->  ( card `  ~P A )  e.  om )
2118, 20sseldi 3191 . . . . 5  |-  ( A  e.  om  ->  ( card `  ~P A )  e.  Fin )
22 php3 7063 . . . . . 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 6906 . . . 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 5555 . . . . . 6  |-  ( F `
 a )  e. 
_V
29 ackbij1lem3 7864 . . . . . . . . 9  |-  ( A  e.  om  ->  A  e.  ( ~P om  i^i  Fin ) )
30 elpwi 3646 . . . . . . . . 9  |-  ( a  e.  ~P A  -> 
a  C_  A )
313ackbij1lem12 7873 . . . . . . . . 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 7871 . . . . . . . . . . 11  |-  F :
( ~P om  i^i  Fin ) --> om
34 peano1 4691 . . . . . . . . . . 11  |-  (/)  e.  om
3533, 34f0cli 5687 . . . . . . . . . 10  |-  ( F `
 a )  e. 
om
36 nnord 4680 . . . . . . . . . 10  |-  ( ( F `  a )  e.  om  ->  Ord  ( F `  a ) )
3735, 36ax-mp 8 . . . . . . . . 9  |-  Ord  ( F `  a )
3833, 34f0cli 5687 . . . . . . . . . 10  |-  ( F `
 A )  e. 
om
39 nnord 4680 . . . . . . . . . 10  |-  ( ( F `  A )  e.  om  ->  Ord  ( F `  A ) )
4038, 39ax-mp 8 . . . . . . . . 9  |-  Ord  ( F `  A )
41 ordsucsssuc 4630 . . . . . . . . 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 7875 . . . . . . . . 9  |-  ( A  e.  om  ->  ( F `  { A } )  =  suc  ( F `  A ) )
453ackbij1lem8 7869 . . . . . . . . 9  |-  ( A  e.  om  ->  ( F `  { A } )  =  (
card `  ~P A ) )
4644, 45eqtr3d 2330 . . . . . . . 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 3227 . . . . . 6  |-  ( ( A  e.  om  /\  a  e.  ~P A
)  ->  suc  ( F `
 a )  C_  ( card `  ~P A ) )
49 sucssel 4501 . . . . . 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 2639 . . . 4  |-  ( A  e.  om  ->  A. a  e.  ~P  A ( F `
 a )  e.  ( card `  ~P A ) )
52 f1fun 5455 . . . . . 6  |-  ( F : ( ~P om  i^i  Fin ) -1-1-> om  ->  Fun 
F )
534, 52ax-mp 8 . . . . 5  |-  Fun  F
54 f1dm 5457 . . . . . . 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 3238 . . . . 5  |-  ( A  e.  om  ->  ~P A  C_  dom  F )
57 funimass4 5589 . . . . 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 3288 . . 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 1632    e. wcel 1696   A.wral 2556   _Vcvv 2801    i^i cin 3164    C_ wss 3165    C. wpss 3166   ~Pcpw 3638   {csn 3653   U_ciun 3921   class class class wbr 4039    e. cmpt 4093   Ord word 4407   Oncon0 4408   suc csuc 4410   omcom 4672    X. cxp 4703   dom cdm 4705   "cima 4708   Fun wfun 5265   -1-1->wf1 5268   ` cfv 5271    ~~ cen 6876    ~< csdm 6878   Fincfn 6879   cardccrd 7584
This theorem is referenced by:  ackbij2lem2  7882
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-recs 6404  df-rdg 6439  df-1o 6495  df-2o 6496  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-cda 7810
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