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Theorem ackbij1lem14 8069
Description: Lemma for ackbij1 8074. (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
ackbij1lem14  |-  ( A  e.  om  ->  ( F `  { A } )  =  suc  ( F `  A ) )
Distinct variable groups:    x, F, y    x, A, y

Proof of Theorem ackbij1lem14
Dummy variables  a 
b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ackbij.f . . 3  |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ y  e.  x  ( {
y }  X.  ~P y ) ) )
21ackbij1lem8 8063 . 2  |-  ( A  e.  om  ->  ( F `  { A } )  =  (
card `  ~P A ) )
3 pweq 3762 . . . . 5  |-  ( a  =  (/)  ->  ~P a  =  ~P (/) )
43fveq2d 5691 . . . 4  |-  ( a  =  (/)  ->  ( card `  ~P a )  =  ( card `  ~P (/) ) )
5 fveq2 5687 . . . . 5  |-  ( a  =  (/)  ->  ( F `
 a )  =  ( F `  (/) ) )
6 suceq 4606 . . . . 5  |-  ( ( F `  a )  =  ( F `  (/) )  ->  suc  ( F `
 a )  =  suc  ( F `  (/) ) )
75, 6syl 16 . . . 4  |-  ( a  =  (/)  ->  suc  ( F `  a )  =  suc  ( F `  (/) ) )
84, 7eqeq12d 2418 . . 3  |-  ( a  =  (/)  ->  ( (
card `  ~P a
)  =  suc  ( F `  a )  <->  (
card `  ~P (/) )  =  suc  ( F `  (/) ) ) )
9 pweq 3762 . . . . 5  |-  ( a  =  b  ->  ~P a  =  ~P b
)
109fveq2d 5691 . . . 4  |-  ( a  =  b  ->  ( card `  ~P a )  =  ( card `  ~P b ) )
11 fveq2 5687 . . . . 5  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
12 suceq 4606 . . . . 5  |-  ( ( F `  a )  =  ( F `  b )  ->  suc  ( F `  a )  =  suc  ( F `
 b ) )
1311, 12syl 16 . . . 4  |-  ( a  =  b  ->  suc  ( F `  a )  =  suc  ( F `
 b ) )
1410, 13eqeq12d 2418 . . 3  |-  ( a  =  b  ->  (
( card `  ~P a
)  =  suc  ( F `  a )  <->  (
card `  ~P b
)  =  suc  ( F `  b )
) )
15 pweq 3762 . . . . 5  |-  ( a  =  suc  b  ->  ~P a  =  ~P suc  b )
1615fveq2d 5691 . . . 4  |-  ( a  =  suc  b  -> 
( card `  ~P a
)  =  ( card `  ~P suc  b ) )
17 fveq2 5687 . . . . 5  |-  ( a  =  suc  b  -> 
( F `  a
)  =  ( F `
 suc  b )
)
18 suceq 4606 . . . . 5  |-  ( ( F `  a )  =  ( F `  suc  b )  ->  suc  ( F `  a )  =  suc  ( F `
 suc  b )
)
1917, 18syl 16 . . . 4  |-  ( a  =  suc  b  ->  suc  ( F `  a
)  =  suc  ( F `  suc  b ) )
2016, 19eqeq12d 2418 . . 3  |-  ( a  =  suc  b  -> 
( ( card `  ~P a )  =  suc  ( F `  a )  <-> 
( card `  ~P suc  b
)  =  suc  ( F `  suc  b ) ) )
21 pweq 3762 . . . . 5  |-  ( a  =  A  ->  ~P a  =  ~P A
)
2221fveq2d 5691 . . . 4  |-  ( a  =  A  ->  ( card `  ~P a )  =  ( card `  ~P A ) )
23 fveq2 5687 . . . . 5  |-  ( a  =  A  ->  ( F `  a )  =  ( F `  A ) )
24 suceq 4606 . . . . 5  |-  ( ( F `  a )  =  ( F `  A )  ->  suc  ( F `  a )  =  suc  ( F `
 A ) )
2523, 24syl 16 . . . 4  |-  ( a  =  A  ->  suc  ( F `  a )  =  suc  ( F `
 A ) )
2622, 25eqeq12d 2418 . . 3  |-  ( a  =  A  ->  (
( card `  ~P a
)  =  suc  ( F `  a )  <->  (
card `  ~P A )  =  suc  ( F `
 A ) ) )
27 df-1o 6683 . . . 4  |-  1o  =  suc  (/)
28 pw0 3905 . . . . . 6  |-  ~P (/)  =  { (/)
}
2928fveq2i 5690 . . . . 5  |-  ( card `  ~P (/) )  =  (
card `  { (/) } )
30 0ex 4299 . . . . . 6  |-  (/)  e.  _V
31 cardsn 7812 . . . . . 6  |-  ( (/)  e.  _V  ->  ( card `  { (/) } )  =  1o )
3230, 31ax-mp 8 . . . . 5  |-  ( card `  { (/) } )  =  1o
3329, 32eqtri 2424 . . . 4  |-  ( card `  ~P (/) )  =  1o
341ackbij1lem13 8068 . . . . 5  |-  ( F `
 (/) )  =  (/)
35 suceq 4606 . . . . 5  |-  ( ( F `  (/) )  =  (/)  ->  suc  ( F `  (/) )  =  suc  (/) )
3634, 35ax-mp 8 . . . 4  |-  suc  ( F `  (/) )  =  suc  (/)
3727, 33, 363eqtr4i 2434 . . 3  |-  ( card `  ~P (/) )  =  suc  ( F `  (/) )
38 oveq2 6048 . . . . . 6  |-  ( (
card `  ~P b
)  =  suc  ( F `  b )  ->  ( ( card `  ~P b )  +o  ( card `  ~P b ) )  =  ( (
card `  ~P b
)  +o  suc  ( F `  b )
) )
3938adantl 453 . . . . 5  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( ( card `  ~P b )  +o  ( card `  ~P b ) )  =  ( ( card `  ~P b )  +o  suc  ( F `  b ) ) )
40 ackbij1lem5 8060 . . . . . 6  |-  ( b  e.  om  ->  ( card `  ~P suc  b
)  =  ( (
card `  ~P b
)  +o  ( card `  ~P b ) ) )
4140adantr 452 . . . . 5  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( card `  ~P suc  b )  =  ( ( card `  ~P b )  +o  ( card `  ~P b ) ) )
42 df-suc 4547 . . . . . . . . . 10  |-  suc  b  =  ( b  u. 
{ b } )
4342equncomi 3453 . . . . . . . . 9  |-  suc  b  =  ( { b }  u.  b )
4443fveq2i 5690 . . . . . . . 8  |-  ( F `
 suc  b )  =  ( F `  ( { b }  u.  b ) )
45 ackbij1lem4 8059 . . . . . . . . . . 11  |-  ( b  e.  om  ->  { b }  e.  ( ~P
om  i^i  Fin )
)
4645adantr 452 . . . . . . . . . 10  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  { b }  e.  ( ~P om  i^i  Fin ) )
47 ackbij1lem3 8058 . . . . . . . . . . 11  |-  ( b  e.  om  ->  b  e.  ( ~P om  i^i  Fin ) )
4847adantr 452 . . . . . . . . . 10  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  b  e.  ( ~P om  i^i  Fin ) )
49 incom 3493 . . . . . . . . . . . 12  |-  ( { b }  i^i  b
)  =  ( b  i^i  { b } )
50 nnord 4812 . . . . . . . . . . . . 13  |-  ( b  e.  om  ->  Ord  b )
51 orddisj 4579 . . . . . . . . . . . . 13  |-  ( Ord  b  ->  ( b  i^i  { b } )  =  (/) )
5250, 51syl 16 . . . . . . . . . . . 12  |-  ( b  e.  om  ->  (
b  i^i  { b } )  =  (/) )
5349, 52syl5eq 2448 . . . . . . . . . . 11  |-  ( b  e.  om  ->  ( { b }  i^i  b )  =  (/) )
5453adantr 452 . . . . . . . . . 10  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( {
b }  i^i  b
)  =  (/) )
551ackbij1lem9 8064 . . . . . . . . . 10  |-  ( ( { b }  e.  ( ~P om  i^i  Fin )  /\  b  e.  ( ~P om  i^i  Fin )  /\  ( { b }  i^i  b )  =  (/) )  ->  ( F `  ( {
b }  u.  b
) )  =  ( ( F `  {
b } )  +o  ( F `  b
) ) )
5646, 48, 54, 55syl3anc 1184 . . . . . . . . 9  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( F `  ( { b }  u.  b ) )  =  ( ( F `
 { b } )  +o  ( F `
 b ) ) )
571ackbij1lem8 8063 . . . . . . . . . . 11  |-  ( b  e.  om  ->  ( F `  { b } )  =  (
card `  ~P b
) )
5857adantr 452 . . . . . . . . . 10  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( F `  { b } )  =  ( card `  ~P b ) )
5958oveq1d 6055 . . . . . . . . 9  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( ( F `  { b } )  +o  ( F `  b )
)  =  ( (
card `  ~P b
)  +o  ( F `
 b ) ) )
6056, 59eqtrd 2436 . . . . . . . 8  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( F `  ( { b }  u.  b ) )  =  ( ( card `  ~P b )  +o  ( F `  b
) ) )
6144, 60syl5eq 2448 . . . . . . 7  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( F `  suc  b )  =  ( ( card `  ~P b )  +o  ( F `  b )
) )
62 suceq 4606 . . . . . . 7  |-  ( ( F `  suc  b
)  =  ( (
card `  ~P b
)  +o  ( F `
 b ) )  ->  suc  ( F `  suc  b )  =  suc  ( ( card `  ~P b )  +o  ( F `  b
) ) )
6361, 62syl 16 . . . . . 6  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  suc  ( F `
 suc  b )  =  suc  ( ( card `  ~P b )  +o  ( F `  b
) ) )
64 nnfi 7258 . . . . . . . . . 10  |-  ( b  e.  om  ->  b  e.  Fin )
65 pwfi 7360 . . . . . . . . . 10  |-  ( b  e.  Fin  <->  ~P b  e.  Fin )
6664, 65sylib 189 . . . . . . . . 9  |-  ( b  e.  om  ->  ~P b  e.  Fin )
6766adantr 452 . . . . . . . 8  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ~P b  e.  Fin )
68 ficardom 7804 . . . . . . . 8  |-  ( ~P b  e.  Fin  ->  (
card `  ~P b
)  e.  om )
6967, 68syl 16 . . . . . . 7  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( card `  ~P b )  e. 
om )
701ackbij1lem10 8065 . . . . . . . . 9  |-  F :
( ~P om  i^i  Fin ) --> om
7170ffvelrni 5828 . . . . . . . 8  |-  ( b  e.  ( ~P om  i^i  Fin )  ->  ( F `  b )  e.  om )
7248, 71syl 16 . . . . . . 7  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( F `  b )  e.  om )
73 nnasuc 6808 . . . . . . 7  |-  ( ( ( card `  ~P b )  e.  om  /\  ( F `  b
)  e.  om )  ->  ( ( card `  ~P b )  +o  suc  ( F `  b ) )  =  suc  (
( card `  ~P b
)  +o  ( F `
 b ) ) )
7469, 72, 73syl2anc 643 . . . . . 6  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( ( card `  ~P b )  +o  suc  ( F `
 b ) )  =  suc  ( (
card `  ~P b
)  +o  ( F `
 b ) ) )
7563, 74eqtr4d 2439 . . . . 5  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  suc  ( F `
 suc  b )  =  ( ( card `  ~P b )  +o 
suc  ( F `  b ) ) )
7639, 41, 753eqtr4d 2446 . . . 4  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( card `  ~P suc  b )  =  suc  ( F `
 suc  b )
)
7776ex 424 . . 3  |-  ( b  e.  om  ->  (
( card `  ~P b
)  =  suc  ( F `  b )  ->  ( card `  ~P suc  b )  =  suc  ( F `  suc  b
) ) )
788, 14, 20, 26, 37, 77finds 4830 . 2  |-  ( A  e.  om  ->  ( card `  ~P A )  =  suc  ( F `
 A ) )
792, 78eqtrd 2436 1  |-  ( A  e.  om  ->  ( F `  { A } )  =  suc  ( F `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649    e. wcel 1721   _Vcvv 2916    u. cun 3278    i^i cin 3279   (/)c0 3588   ~Pcpw 3759   {csn 3774   U_ciun 4053    e. cmpt 4226   Ord word 4540   suc csuc 4543   omcom 4804    X. cxp 4835   ` cfv 5413  (class class class)co 6040   1oc1o 6676    +o coa 6680   Fincfn 7068   cardccrd 7778
This theorem is referenced by:  ackbij1lem15  8070  ackbij1lem18  8073  ackbij1b  8075
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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