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Theorem ackbij1lem14 7875
Description: Lemma for ackbij1 7880. (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 7869 . 2  |-  ( A  e.  om  ->  ( F `  { A } )  =  (
card `  ~P A ) )
3 pweq 3641 . . . . 5  |-  ( a  =  (/)  ->  ~P a  =  ~P (/) )
43fveq2d 5545 . . . 4  |-  ( a  =  (/)  ->  ( card `  ~P a )  =  ( card `  ~P (/) ) )
5 fveq2 5541 . . . . 5  |-  ( a  =  (/)  ->  ( F `
 a )  =  ( F `  (/) ) )
6 suceq 4473 . . . . 5  |-  ( ( F `  a )  =  ( F `  (/) )  ->  suc  ( F `
 a )  =  suc  ( F `  (/) ) )
75, 6syl 15 . . . 4  |-  ( a  =  (/)  ->  suc  ( F `  a )  =  suc  ( F `  (/) ) )
84, 7eqeq12d 2310 . . 3  |-  ( a  =  (/)  ->  ( (
card `  ~P a
)  =  suc  ( F `  a )  <->  (
card `  ~P (/) )  =  suc  ( F `  (/) ) ) )
9 pweq 3641 . . . . 5  |-  ( a  =  b  ->  ~P a  =  ~P b
)
109fveq2d 5545 . . . 4  |-  ( a  =  b  ->  ( card `  ~P a )  =  ( card `  ~P b ) )
11 fveq2 5541 . . . . 5  |-  ( a  =  b  ->  ( F `  a )  =  ( F `  b ) )
12 suceq 4473 . . . . 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 2310 . . 3  |-  ( a  =  b  ->  (
( card `  ~P a
)  =  suc  ( F `  a )  <->  (
card `  ~P b
)  =  suc  ( F `  b )
) )
15 pweq 3641 . . . . 5  |-  ( a  =  suc  b  ->  ~P a  =  ~P suc  b )
1615fveq2d 5545 . . . 4  |-  ( a  =  suc  b  -> 
( card `  ~P a
)  =  ( card `  ~P suc  b ) )
17 fveq2 5541 . . . . 5  |-  ( a  =  suc  b  -> 
( F `  a
)  =  ( F `
 suc  b )
)
18 suceq 4473 . . . . 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 2310 . . 3  |-  ( a  =  suc  b  -> 
( ( card `  ~P a )  =  suc  ( F `  a )  <-> 
( card `  ~P suc  b
)  =  suc  ( F `  suc  b ) ) )
21 pweq 3641 . . . . 5  |-  ( a  =  A  ->  ~P a  =  ~P A
)
2221fveq2d 5545 . . . 4  |-  ( a  =  A  ->  ( card `  ~P a )  =  ( card `  ~P A ) )
23 fveq2 5541 . . . . 5  |-  ( a  =  A  ->  ( F `  a )  =  ( F `  A ) )
24 suceq 4473 . . . . 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 2310 . . 3  |-  ( a  =  A  ->  (
( card `  ~P a
)  =  suc  ( F `  a )  <->  (
card `  ~P A )  =  suc  ( F `
 A ) ) )
27 df-1o 6495 . . . 4  |-  1o  =  suc  (/)
28 pw0 3778 . . . . . 6  |-  ~P (/)  =  { (/)
}
2928fveq2i 5544 . . . . 5  |-  ( card `  ~P (/) )  =  (
card `  { (/) } )
30 0ex 4166 . . . . . 6  |-  (/)  e.  _V
31 cardsn 7618 . . . . . 6  |-  ( (/)  e.  _V  ->  ( card `  { (/) } )  =  1o )
3230, 31ax-mp 8 . . . . 5  |-  ( card `  { (/) } )  =  1o
3329, 32eqtri 2316 . . . 4  |-  ( card `  ~P (/) )  =  1o
341ackbij1lem13 7874 . . . . 5  |-  ( F `
 (/) )  =  (/)
35 suceq 4473 . . . . 5  |-  ( ( F `  (/) )  =  (/)  ->  suc  ( F `  (/) )  =  suc  (/) )
3634, 35ax-mp 8 . . . 4  |-  suc  ( F `  (/) )  =  suc  (/)
3727, 33, 363eqtr4i 2326 . . 3  |-  ( card `  ~P (/) )  =  suc  ( F `  (/) )
38 oveq2 5882 . . . . . 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 7866 . . . . . 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 4414 . . . . . . . . . 10  |-  suc  b  =  ( b  u. 
{ b } )
4342equncomi 3334 . . . . . . . . 9  |-  suc  b  =  ( { b }  u.  b )
4443fveq2i 5544 . . . . . . . 8  |-  ( F `
 suc  b )  =  ( F `  ( { b }  u.  b ) )
45 ackbij1lem4 7865 . . . . . . . . . . 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 7864 . . . . . . . . . . 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 3374 . . . . . . . . . . . 12  |-  ( { b }  i^i  b
)  =  ( b  i^i  { b } )
50 nnord 4680 . . . . . . . . . . . . 13  |-  ( b  e.  om  ->  Ord  b )
51 orddisj 4446 . . . . . . . . . . . . 13  |-  ( Ord  b  ->  ( b  i^i  { b } )  =  (/) )
5250, 51syl 15 . . . . . . . . . . . 12  |-  ( b  e.  om  ->  (
b  i^i  { b } )  =  (/) )
5349, 52syl5eq 2340 . . . . . . . . . . 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 7870 . . . . . . . . . 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 7869 . . . . . . . . . . 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 5889 . . . . . . . . 9  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( ( F `  { b } )  +o  ( F `  b )
)  =  ( (
card `  ~P b
)  +o  ( F `
 b ) ) )
6056, 59eqtrd 2328 . . . . . . . 8  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( F `  ( { b }  u.  b ) )  =  ( ( card `  ~P b )  +o  ( F `  b
) ) )
6144, 60syl5eq 2340 . . . . . . 7  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  ( F `  suc  b )  =  ( ( card `  ~P b )  +o  ( F `  b )
) )
62 suceq 4473 . . . . . . 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 7069 . . . . . . . . . 10  |-  ( b  e.  om  ->  b  e.  Fin )
65 pwfi 7167 . . . . . . . . . 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 7610 . . . . . . . 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 7871 . . . . . . . . 9  |-  F :
( ~P om  i^i  Fin ) --> om
7170ffvelrni 5680 . . . . . . . 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 6620 . . . . . . 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 2331 . . . . 5  |-  ( ( b  e.  om  /\  ( card `  ~P b
)  =  suc  ( F `  b )
)  ->  suc  ( F `
 suc  b )  =  ( ( card `  ~P b )  +o 
suc  ( F `  b ) ) )
7639, 41, 753eqtr4d 2338 . . . 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 4698 . 2  |-  ( A  e.  om  ->  ( card `  ~P A )  =  suc  ( F `
 A ) )
792, 78eqtrd 2328 1  |-  ( A  e.  om  ->  ( F `  { A } )  =  suc  ( F `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    u. cun 3163    i^i cin 3164   (/)c0 3468   ~Pcpw 3638   {csn 3653   U_ciun 3921    e. cmpt 4093   Ord word 4407   suc csuc 4410   omcom 4672    X. cxp 4703   ` cfv 5271  (class class class)co 5874   1oc1o 6488    +o coa 6492   Fincfn 6879   cardccrd 7584
This theorem is referenced by:  ackbij1lem15  7876  ackbij1lem18  7879  ackbij1b  7881
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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