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Theorem ackbij1lem8 8063
Description: Lemma for ackbij1 8074. (Contributed by Stefan O'Rear, 19-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
ackbij1lem8  |-  ( A  e.  om  ->  ( F `  { A } )  =  (
card `  ~P A ) )
Distinct variable groups:    x, F, y    x, A, y

Proof of Theorem ackbij1lem8
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 sneq 3785 . . . 4  |-  ( a  =  A  ->  { a }  =  { A } )
21fveq2d 5691 . . 3  |-  ( a  =  A  ->  ( F `  { a } )  =  ( F `  { A } ) )
3 pweq 3762 . . . 4  |-  ( a  =  A  ->  ~P a  =  ~P A
)
43fveq2d 5691 . . 3  |-  ( a  =  A  ->  ( card `  ~P a )  =  ( card `  ~P A ) )
52, 4eqeq12d 2418 . 2  |-  ( a  =  A  ->  (
( F `  {
a } )  =  ( card `  ~P a )  <->  ( F `  { A } )  =  ( card `  ~P A ) ) )
6 ackbij1lem4 8059 . . . 4  |-  ( a  e.  om  ->  { a }  e.  ( ~P
om  i^i  Fin )
)
7 ackbij.f . . . . 5  |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ y  e.  x  ( {
y }  X.  ~P y ) ) )
87ackbij1lem7 8062 . . . 4  |-  ( { a }  e.  ( ~P om  i^i  Fin )  ->  ( F `  { a } )  =  ( card `  U_ y  e.  { a }  ( { y }  X.  ~P y ) ) )
96, 8syl 16 . . 3  |-  ( a  e.  om  ->  ( F `  { a } )  =  (
card `  U_ y  e. 
{ a }  ( { y }  X.  ~P y ) ) )
10 vex 2919 . . . . . 6  |-  a  e. 
_V
11 sneq 3785 . . . . . . 7  |-  ( y  =  a  ->  { y }  =  { a } )
12 pweq 3762 . . . . . . 7  |-  ( y  =  a  ->  ~P y  =  ~P a
)
1311, 12xpeq12d 4862 . . . . . 6  |-  ( y  =  a  ->  ( { y }  X.  ~P y )  =  ( { a }  X.  ~P a ) )
1410, 13iunxsn 4130 . . . . 5  |-  U_ y  e.  { a }  ( { y }  X.  ~P y )  =  ( { a }  X.  ~P a )
1514fveq2i 5690 . . . 4  |-  ( card `  U_ y  e.  {
a }  ( { y }  X.  ~P y ) )  =  ( card `  ( { a }  X.  ~P a ) )
1610pwex 4342 . . . . . 6  |-  ~P a  e.  _V
17 xpsnen2g 7160 . . . . . 6  |-  ( ( a  e.  _V  /\  ~P a  e.  _V )  ->  ( { a }  X.  ~P a
)  ~~  ~P a
)
1810, 16, 17mp2an 654 . . . . 5  |-  ( { a }  X.  ~P a )  ~~  ~P a
19 carden2b 7810 . . . . 5  |-  ( ( { a }  X.  ~P a )  ~~  ~P a  ->  ( card `  ( { a }  X.  ~P a ) )  =  ( card `  ~P a ) )
2018, 19ax-mp 8 . . . 4  |-  ( card `  ( { a }  X.  ~P a ) )  =  ( card `  ~P a )
2115, 20eqtri 2424 . . 3  |-  ( card `  U_ y  e.  {
a }  ( { y }  X.  ~P y ) )  =  ( card `  ~P a )
229, 21syl6eq 2452 . 2  |-  ( a  e.  om  ->  ( F `  { a } )  =  (
card `  ~P a
) )
235, 22vtoclga 2977 1  |-  ( A  e.  om  ->  ( F `  { A } )  =  (
card `  ~P A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   _Vcvv 2916    i^i cin 3279   ~Pcpw 3759   {csn 3774   U_ciun 4053   class class class wbr 4172    e. cmpt 4226   omcom 4804    X. cxp 4835   ` cfv 5413    ~~ cen 7065   Fincfn 7068   cardccrd 7778
This theorem is referenced by:  ackbij1lem14  8069  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-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-rab 2675  df-v 2918  df-sbc 3122  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-1st 6308  df-2nd 6309  df-1o 6683  df-er 6864  df-en 7069  df-fin 7072  df-card 7782
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