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Theorem ackbij1lem8 8071
Description: Lemma for ackbij1 8082. (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 3793 . . . 4  |-  ( a  =  A  ->  { a }  =  { A } )
21fveq2d 5699 . . 3  |-  ( a  =  A  ->  ( F `  { a } )  =  ( F `  { A } ) )
3 pweq 3770 . . . 4  |-  ( a  =  A  ->  ~P a  =  ~P A
)
43fveq2d 5699 . . 3  |-  ( a  =  A  ->  ( card `  ~P a )  =  ( card `  ~P A ) )
52, 4eqeq12d 2426 . 2  |-  ( a  =  A  ->  (
( F `  {
a } )  =  ( card `  ~P a )  <->  ( F `  { A } )  =  ( card `  ~P A ) ) )
6 ackbij1lem4 8067 . . . 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 8070 . . . 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 2927 . . . . . 6  |-  a  e. 
_V
11 sneq 3793 . . . . . . 7  |-  ( y  =  a  ->  { y }  =  { a } )
12 pweq 3770 . . . . . . 7  |-  ( y  =  a  ->  ~P y  =  ~P a
)
1311, 12xpeq12d 4870 . . . . . 6  |-  ( y  =  a  ->  ( { y }  X.  ~P y )  =  ( { a }  X.  ~P a ) )
1410, 13iunxsn 4138 . . . . 5  |-  U_ y  e.  { a }  ( { y }  X.  ~P y )  =  ( { a }  X.  ~P a )
1514fveq2i 5698 . . . 4  |-  ( card `  U_ y  e.  {
a }  ( { y }  X.  ~P y ) )  =  ( card `  ( { a }  X.  ~P a ) )
1610pwex 4350 . . . . . 6  |-  ~P a  e.  _V
17 xpsnen2g 7168 . . . . . 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 7818 . . . . 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 2432 . . 3  |-  ( card `  U_ y  e.  {
a }  ( { y }  X.  ~P y ) )  =  ( card `  ~P a )
229, 21syl6eq 2460 . 2  |-  ( a  e.  om  ->  ( F `  { a } )  =  (
card `  ~P a
) )
235, 22vtoclga 2985 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 2924    i^i cin 3287   ~Pcpw 3767   {csn 3782   U_ciun 4061   class class class wbr 4180    e. cmpt 4234   omcom 4812    X. cxp 4843   ` cfv 5421    ~~ cen 7073   Fincfn 7076   cardccrd 7786
This theorem is referenced by:  ackbij1lem14  8077  ackbij1b  8083
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 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
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 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-pss 3304  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-tp 3790  df-op 3791  df-uni 3984  df-int 4019  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-tr 4271  df-eprel 4462  df-id 4466  df-po 4471  df-so 4472  df-fr 4509  df-we 4511  df-ord 4552  df-on 4553  df-lim 4554  df-suc 4555  df-om 4813  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-1st 6316  df-2nd 6317  df-1o 6691  df-er 6872  df-en 7077  df-fin 7080  df-card 7790
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