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Theorem ackbij1lem8 7853
Description: Lemma for ackbij1 7864. (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 3651 . . . 4  |-  ( a  =  A  ->  { a }  =  { A } )
21fveq2d 5529 . . 3  |-  ( a  =  A  ->  ( F `  { a } )  =  ( F `  { A } ) )
3 pweq 3628 . . . 4  |-  ( a  =  A  ->  ~P a  =  ~P A
)
43fveq2d 5529 . . 3  |-  ( a  =  A  ->  ( card `  ~P a )  =  ( card `  ~P A ) )
52, 4eqeq12d 2297 . 2  |-  ( a  =  A  ->  (
( F `  {
a } )  =  ( card `  ~P a )  <->  ( F `  { A } )  =  ( card `  ~P A ) ) )
6 ackbij1lem4 7849 . . . 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 7852 . . . 4  |-  ( { a }  e.  ( ~P om  i^i  Fin )  ->  ( F `  { a } )  =  ( card `  U_ y  e.  { a }  ( { y }  X.  ~P y ) ) )
96, 8syl 15 . . 3  |-  ( a  e.  om  ->  ( F `  { a } )  =  (
card `  U_ y  e. 
{ a }  ( { y }  X.  ~P y ) ) )
10 vex 2791 . . . . . 6  |-  a  e. 
_V
11 sneq 3651 . . . . . . 7  |-  ( y  =  a  ->  { y }  =  { a } )
12 pweq 3628 . . . . . . 7  |-  ( y  =  a  ->  ~P y  =  ~P a
)
1311, 12xpeq12d 4714 . . . . . 6  |-  ( y  =  a  ->  ( { y }  X.  ~P y )  =  ( { a }  X.  ~P a ) )
1410, 13iunxsn 3981 . . . . 5  |-  U_ y  e.  { a }  ( { y }  X.  ~P y )  =  ( { a }  X.  ~P a )
1514fveq2i 5528 . . . 4  |-  ( card `  U_ y  e.  {
a }  ( { y }  X.  ~P y ) )  =  ( card `  ( { a }  X.  ~P a ) )
1610pwex 4193 . . . . . 6  |-  ~P a  e.  _V
17 xpsnen2g 6955 . . . . . 6  |-  ( ( a  e.  _V  /\  ~P a  e.  _V )  ->  ( { a }  X.  ~P a
)  ~~  ~P a
)
1810, 16, 17mp2an 653 . . . . 5  |-  ( { a }  X.  ~P a )  ~~  ~P a
19 carden2b 7600 . . . . 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 2303 . . 3  |-  ( card `  U_ y  e.  {
a }  ( { y }  X.  ~P y ) )  =  ( card `  ~P a )
229, 21syl6eq 2331 . 2  |-  ( a  e.  om  ->  ( F `  { a } )  =  (
card `  ~P a
) )
235, 22vtoclga 2849 1  |-  ( A  e.  om  ->  ( F `  { A } )  =  (
card `  ~P A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   ~Pcpw 3625   {csn 3640   U_ciun 3905   class class class wbr 4023    e. cmpt 4077   omcom 4656    X. cxp 4687   ` cfv 5255    ~~ cen 6860   Fincfn 6863   cardccrd 7568
This theorem is referenced by:  ackbij1lem14  7859  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-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-rab 2552  df-v 2790  df-sbc 2992  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-1st 6122  df-2nd 6123  df-1o 6479  df-er 6660  df-en 6864  df-fin 6867  df-card 7572
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