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Theorem ackbij1lem14 8113
Description: Lemma for ackbij1 8118. (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 8107 . 2  |-  ( A  e.  om  ->  ( F `  { A } )  =  (
card `  ~P A ) )
3 pweq 3802 . . . . 5  |-  ( a  =  (/)  ->  ~P a  =  ~P (/) )
43fveq2d 5732 . . . 4  |-  ( a  =  (/)  ->  ( card `  ~P a )  =  ( card `  ~P (/) ) )
5 fveq2 5728 . . . . 5  |-  ( a  =  (/)  ->  ( F `
 a )  =  ( F `  (/) ) )
6 suceq 4646 . . . . 5  |-  ( ( F `  a )  =  ( F `  (/) )  ->  suc  ( F `
 a )  =  suc  ( F `  (/) ) )
75, 6syl 16 . . . 4  |-  ( a  =  (/)  ->  suc  ( F `  a )  =  suc  ( F `  (/) ) )
84, 7eqeq12d 2450 . . 3  |-  ( a  =  (/)  ->  ( (
card `  ~P a
)  =  suc  ( F `  a )  <->  (
card `  ~P (/) )  =  suc  ( F `  (/) ) ) )
9 pweq 3802 . . . . 5  |-  ( a  =  b  ->  ~P a  =  ~P b
)
109fveq2d 5732 . . . 4  |-  ( a  =  b  ->  ( card `  ~P a )  =  ( card `  ~P b ) )
11 fveq2 5728 . . . . 5  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
12 suceq 4646 . . . . 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 2450 . . 3  |-  ( a  =  b  ->  (
( card `  ~P a
)  =  suc  ( F `  a )  <->  (
card `  ~P b
)  =  suc  ( F `  b )
) )
15 pweq 3802 . . . . 5  |-  ( a  =  suc  b  ->  ~P a  =  ~P suc  b )
1615fveq2d 5732 . . . 4  |-  ( a  =  suc  b  -> 
( card `  ~P a
)  =  ( card `  ~P suc  b ) )
17 fveq2 5728 . . . . 5  |-  ( a  =  suc  b  -> 
( F `  a
)  =  ( F `
 suc  b )
)
18 suceq 4646 . . . . 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 2450 . . 3  |-  ( a  =  suc  b  -> 
( ( card `  ~P a )  =  suc  ( F `  a )  <-> 
( card `  ~P suc  b
)  =  suc  ( F `  suc  b ) ) )
21 pweq 3802 . . . . 5  |-  ( a  =  A  ->  ~P a  =  ~P A
)
2221fveq2d 5732 . . . 4  |-  ( a  =  A  ->  ( card `  ~P a )  =  ( card `  ~P A ) )
23 fveq2 5728 . . . . 5  |-  ( a  =  A  ->  ( F `  a )  =  ( F `  A ) )
24 suceq 4646 . . . . 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 2450 . . 3  |-  ( a  =  A  ->  (
( card `  ~P a
)  =  suc  ( F `  a )  <->  (
card `  ~P A )  =  suc  ( F `
 A ) ) )
27 df-1o 6724 . . . 4  |-  1o  =  suc  (/)
28 pw0 3945 . . . . . 6  |-  ~P (/)  =  { (/)
}
2928fveq2i 5731 . . . . 5  |-  ( card `  ~P (/) )  =  (
card `  { (/) } )
30 0ex 4339 . . . . . 6  |-  (/)  e.  _V
31 cardsn 7856 . . . . . 6  |-  ( (/)  e.  _V  ->  ( card `  { (/) } )  =  1o )
3230, 31ax-mp 8 . . . . 5  |-  ( card `  { (/) } )  =  1o
3329, 32eqtri 2456 . . . 4  |-  ( card `  ~P (/) )  =  1o
341ackbij1lem13 8112 . . . . 5  |-  ( F `
 (/) )  =  (/)
35 suceq 4646 . . . . 5  |-  ( ( F `  (/) )  =  (/)  ->  suc  ( F `  (/) )  =  suc  (/) )
3634, 35ax-mp 8 . . . 4  |-  suc  ( F `  (/) )  =  suc  (/)
3727, 33, 363eqtr4i 2466 . . 3  |-  ( card `  ~P (/) )  =  suc  ( F `  (/) )
38 oveq2 6089 . . . . . 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 8104 . . . . . 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 4587 . . . . . . . . . 10  |-  suc  b  =  ( b  u. 
{ b } )
4342equncomi 3493 . . . . . . . . 9  |-  suc  b  =  ( { b }  u.  b )
4443fveq2i 5731 . . . . . . . 8  |-  ( F `
 suc  b )  =  ( F `  ( { b }  u.  b ) )
45 ackbij1lem4 8103 . . . . . . . . . . 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 8102 . . . . . . . . . . 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 3533 . . . . . . . . . . . 12  |-  ( { b }  i^i  b
)  =  ( b  i^i  { b } )
50 nnord 4853 . . . . . . . . . . . . 13  |-  ( b  e.  om  ->  Ord  b )
51 orddisj 4619 . . . . . . . . . . . . 13  |-  ( Ord  b  ->  ( b  i^i  { b } )  =  (/) )
5250, 51syl 16 . . . . . . . . . . . 12  |-  ( b  e.  om  ->  (
b  i^i  { b } )  =  (/) )
5349, 52syl5eq 2480 . . . . . . . . . . 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 8108 . . . . . . . . . 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 8107 . . . . . . . . . . 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 6096 . . . . . . . . 9  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( ( F `  { b } )  +o  ( F `  b )
)  =  ( (
card `  ~P b
)  +o  ( F `
 b ) ) )
6056, 59eqtrd 2468 . . . . . . . 8  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( F `  ( { b }  u.  b ) )  =  ( ( card `  ~P b )  +o  ( F `  b
) ) )
6144, 60syl5eq 2480 . . . . . . 7  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( F `  suc  b )  =  ( ( card `  ~P b )  +o  ( F `  b )
) )
62 suceq 4646 . . . . . . 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 7299 . . . . . . . . . 10  |-  ( b  e.  om  ->  b  e.  Fin )
65 pwfi 7402 . . . . . . . . . 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 7848 . . . . . . . 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 8109 . . . . . . . . 9  |-  F :
( ~P om  i^i  Fin ) --> om
7170ffvelrni 5869 . . . . . . . 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 6849 . . . . . . 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 2471 . . . . 5  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  suc  ( F `
 suc  b )  =  ( ( card `  ~P b )  +o 
suc  ( F `  b ) ) )
7639, 41, 753eqtr4d 2478 . . . 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 4871 . 2  |-  ( A  e.  om  ->  ( card `  ~P A )  =  suc  ( F `
 A ) )
792, 78eqtrd 2468 1  |-  ( A  e.  om  ->  ( F `  { A } )  =  suc  ( F `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   _Vcvv 2956    u. cun 3318    i^i cin 3319   (/)c0 3628   ~Pcpw 3799   {csn 3814   U_ciun 4093    e. cmpt 4266   Ord word 4580   suc csuc 4583   omcom 4845    X. cxp 4876   ` cfv 5454  (class class class)co 6081   1oc1o 6717    +o coa 6721   Fincfn 7109   cardccrd 7822
This theorem is referenced by:  ackbij1lem15  8114  ackbij1lem18  8117  ackbij1b  8119
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-recs 6633  df-rdg 6668  df-1o 6724  df-2o 6725  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826  df-cda 8048
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