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Theorem ackbij1lem14 7859
Description: Lemma for ackbij1 7864. (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 7853 . 2  |-  ( A  e.  om  ->  ( F `  { A } )  =  (
card `  ~P A ) )
3 pweq 3628 . . . . 5  |-  ( a  =  (/)  ->  ~P a  =  ~P (/) )
43fveq2d 5529 . . . 4  |-  ( a  =  (/)  ->  ( card `  ~P a )  =  ( card `  ~P (/) ) )
5 fveq2 5525 . . . . 5  |-  ( a  =  (/)  ->  ( F `
 a )  =  ( F `  (/) ) )
6 suceq 4457 . . . . 5  |-  ( ( F `  a )  =  ( F `  (/) )  ->  suc  ( F `
 a )  =  suc  ( F `  (/) ) )
75, 6syl 15 . . . 4  |-  ( a  =  (/)  ->  suc  ( F `  a )  =  suc  ( F `  (/) ) )
84, 7eqeq12d 2297 . . 3  |-  ( a  =  (/)  ->  ( (
card `  ~P a
)  =  suc  ( F `  a )  <->  (
card `  ~P (/) )  =  suc  ( F `  (/) ) ) )
9 pweq 3628 . . . . 5  |-  ( a  =  b  ->  ~P a  =  ~P b
)
109fveq2d 5529 . . . 4  |-  ( a  =  b  ->  ( card `  ~P a )  =  ( card `  ~P b ) )
11 fveq2 5525 . . . . 5  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
12 suceq 4457 . . . . 5  |-  ( ( F `  a )  =  ( F `  b )  ->  suc  ( F `  a )  =  suc  ( F `
 b ) )
1311, 12syl 15 . . . 4  |-  ( a  =  b  ->  suc  ( F `  a )  =  suc  ( F `
 b ) )
1410, 13eqeq12d 2297 . . 3  |-  ( a  =  b  ->  (
( card `  ~P a
)  =  suc  ( F `  a )  <->  (
card `  ~P b
)  =  suc  ( F `  b )
) )
15 pweq 3628 . . . . 5  |-  ( a  =  suc  b  ->  ~P a  =  ~P suc  b )
1615fveq2d 5529 . . . 4  |-  ( a  =  suc  b  -> 
( card `  ~P a
)  =  ( card `  ~P suc  b ) )
17 fveq2 5525 . . . . 5  |-  ( a  =  suc  b  -> 
( F `  a
)  =  ( F `
 suc  b )
)
18 suceq 4457 . . . . 5  |-  ( ( F `  a )  =  ( F `  suc  b )  ->  suc  ( F `  a )  =  suc  ( F `
 suc  b )
)
1917, 18syl 15 . . . 4  |-  ( a  =  suc  b  ->  suc  ( F `  a
)  =  suc  ( F `  suc  b ) )
2016, 19eqeq12d 2297 . . 3  |-  ( a  =  suc  b  -> 
( ( card `  ~P a )  =  suc  ( F `  a )  <-> 
( card `  ~P suc  b
)  =  suc  ( F `  suc  b ) ) )
21 pweq 3628 . . . . 5  |-  ( a  =  A  ->  ~P a  =  ~P A
)
2221fveq2d 5529 . . . 4  |-  ( a  =  A  ->  ( card `  ~P a )  =  ( card `  ~P A ) )
23 fveq2 5525 . . . . 5  |-  ( a  =  A  ->  ( F `  a )  =  ( F `  A ) )
24 suceq 4457 . . . . 5  |-  ( ( F `  a )  =  ( F `  A )  ->  suc  ( F `  a )  =  suc  ( F `
 A ) )
2523, 24syl 15 . . . 4  |-  ( a  =  A  ->  suc  ( F `  a )  =  suc  ( F `
 A ) )
2622, 25eqeq12d 2297 . . 3  |-  ( a  =  A  ->  (
( card `  ~P a
)  =  suc  ( F `  a )  <->  (
card `  ~P A )  =  suc  ( F `
 A ) ) )
27 df-1o 6479 . . . 4  |-  1o  =  suc  (/)
28 pw0 3762 . . . . . 6  |-  ~P (/)  =  { (/)
}
2928fveq2i 5528 . . . . 5  |-  ( card `  ~P (/) )  =  (
card `  { (/) } )
30 0ex 4150 . . . . . 6  |-  (/)  e.  _V
31 cardsn 7602 . . . . . 6  |-  ( (/)  e.  _V  ->  ( card `  { (/) } )  =  1o )
3230, 31ax-mp 8 . . . . 5  |-  ( card `  { (/) } )  =  1o
3329, 32eqtri 2303 . . . 4  |-  ( card `  ~P (/) )  =  1o
341ackbij1lem13 7858 . . . . 5  |-  ( F `
 (/) )  =  (/)
35 suceq 4457 . . . . 5  |-  ( ( F `  (/) )  =  (/)  ->  suc  ( F `  (/) )  =  suc  (/) )
3634, 35ax-mp 8 . . . 4  |-  suc  ( F `  (/) )  =  suc  (/)
3727, 33, 363eqtr4i 2313 . . 3  |-  ( card `  ~P (/) )  =  suc  ( F `  (/) )
38 oveq2 5866 . . . . . 6  |-  ( (
card `  ~P b
)  =  suc  ( F `  b )  ->  ( ( card `  ~P b )  +o  ( card `  ~P b ) )  =  ( (
card `  ~P b
)  +o  suc  ( F `  b )
) )
3938adantl 452 . . . . 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 7850 . . . . . 6  |-  ( b  e.  om  ->  ( card `  ~P suc  b
)  =  ( (
card `  ~P b
)  +o  ( card `  ~P b ) ) )
4140adantr 451 . . . . 5  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( card `  ~P suc  b )  =  ( ( card `  ~P b )  +o  ( card `  ~P b ) ) )
42 df-suc 4398 . . . . . . . . . 10  |-  suc  b  =  ( b  u. 
{ b } )
4342equncomi 3321 . . . . . . . . 9  |-  suc  b  =  ( { b }  u.  b )
4443fveq2i 5528 . . . . . . . 8  |-  ( F `
 suc  b )  =  ( F `  ( { b }  u.  b ) )
45 ackbij1lem4 7849 . . . . . . . . . . 11  |-  ( b  e.  om  ->  { b }  e.  ( ~P
om  i^i  Fin )
)
4645adantr 451 . . . . . . . . . 10  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  { b }  e.  ( ~P om  i^i  Fin ) )
47 ackbij1lem3 7848 . . . . . . . . . . 11  |-  ( b  e.  om  ->  b  e.  ( ~P om  i^i  Fin ) )
4847adantr 451 . . . . . . . . . 10  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  b  e.  ( ~P om  i^i  Fin ) )
49 incom 3361 . . . . . . . . . . . 12  |-  ( { b }  i^i  b
)  =  ( b  i^i  { b } )
50 nnord 4664 . . . . . . . . . . . . 13  |-  ( b  e.  om  ->  Ord  b )
51 orddisj 4430 . . . . . . . . . . . . 13  |-  ( Ord  b  ->  ( b  i^i  { b } )  =  (/) )
5250, 51syl 15 . . . . . . . . . . . 12  |-  ( b  e.  om  ->  (
b  i^i  { b } )  =  (/) )
5349, 52syl5eq 2327 . . . . . . . . . . 11  |-  ( b  e.  om  ->  ( { b }  i^i  b )  =  (/) )
5453adantr 451 . . . . . . . . . 10  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( {
b }  i^i  b
)  =  (/) )
551ackbij1lem9 7854 . . . . . . . . . 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 1182 . . . . . . . . 9  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( F `  ( { b }  u.  b ) )  =  ( ( F `
 { b } )  +o  ( F `
 b ) ) )
571ackbij1lem8 7853 . . . . . . . . . . 11  |-  ( b  e.  om  ->  ( F `  { b } )  =  (
card `  ~P b
) )
5857adantr 451 . . . . . . . . . 10  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( F `  { b } )  =  ( card `  ~P b ) )
5958oveq1d 5873 . . . . . . . . 9  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( ( F `  { b } )  +o  ( F `  b )
)  =  ( (
card `  ~P b
)  +o  ( F `
 b ) ) )
6056, 59eqtrd 2315 . . . . . . . 8  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( F `  ( { b }  u.  b ) )  =  ( ( card `  ~P b )  +o  ( F `  b
) ) )
6144, 60syl5eq 2327 . . . . . . 7  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( F `  suc  b )  =  ( ( card `  ~P b )  +o  ( F `  b )
) )
62 suceq 4457 . . . . . . 7  |-  ( ( F `  suc  b
)  =  ( (
card `  ~P b
)  +o  ( F `
 b ) )  ->  suc  ( F `  suc  b )  =  suc  ( ( card `  ~P b )  +o  ( F `  b
) ) )
6361, 62syl 15 . . . . . 6  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  suc  ( F `
 suc  b )  =  suc  ( ( card `  ~P b )  +o  ( F `  b
) ) )
64 nnfi 7053 . . . . . . . . . 10  |-  ( b  e.  om  ->  b  e.  Fin )
65 pwfi 7151 . . . . . . . . . 10  |-  ( b  e.  Fin  <->  ~P b  e.  Fin )
6664, 65sylib 188 . . . . . . . . 9  |-  ( b  e.  om  ->  ~P b  e.  Fin )
6766adantr 451 . . . . . . . 8  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ~P b  e.  Fin )
68 ficardom 7594 . . . . . . . 8  |-  ( ~P b  e.  Fin  ->  (
card `  ~P b
)  e.  om )
6967, 68syl 15 . . . . . . 7  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( card `  ~P b )  e. 
om )
701ackbij1lem10 7855 . . . . . . . . 9  |-  F :
( ~P om  i^i  Fin ) --> om
7170ffvelrni 5664 . . . . . . . 8  |-  ( b  e.  ( ~P om  i^i  Fin )  ->  ( F `  b )  e.  om )
7248, 71syl 15 . . . . . . 7  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( F `  b )  e.  om )
73 nnasuc 6604 . . . . . . 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 642 . . . . . 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 2318 . . . . 5  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  suc  ( F `
 suc  b )  =  ( ( card `  ~P b )  +o 
suc  ( F `  b ) ) )
7639, 41, 753eqtr4d 2325 . . . 4  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( card `  ~P suc  b )  =  suc  ( F `
 suc  b )
)
7776ex 423 . . 3  |-  ( b  e.  om  ->  (
( card `  ~P b
)  =  suc  ( F `  b )  ->  ( card `  ~P suc  b )  =  suc  ( F `  suc  b
) ) )
788, 14, 20, 26, 37, 77finds 4682 . 2  |-  ( A  e.  om  ->  ( card `  ~P A )  =  suc  ( F `
 A ) )
792, 78eqtrd 2315 1  |-  ( A  e.  om  ->  ( F `  { A } )  =  suc  ( F `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150    i^i cin 3151   (/)c0 3455   ~Pcpw 3625   {csn 3640   U_ciun 3905    e. cmpt 4077   Ord word 4391   suc csuc 4394   omcom 4656    X. cxp 4687   ` cfv 5255  (class class class)co 5858   1oc1o 6472    +o coa 6476   Fincfn 6863   cardccrd 7568
This theorem is referenced by:  ackbij1lem15  7860  ackbij1lem18  7863  ackbij1b  7865
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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