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Theorem ackbij1lem8 8138
Description: Lemma for ackbij1 8149. (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 3849 . . . 4  |-  ( a  =  A  ->  { a }  =  { A } )
21fveq2d 5761 . . 3  |-  ( a  =  A  ->  ( F `  { a } )  =  ( F `  { A } ) )
3 pweq 3826 . . . 4  |-  ( a  =  A  ->  ~P a  =  ~P A
)
43fveq2d 5761 . . 3  |-  ( a  =  A  ->  ( card `  ~P a )  =  ( card `  ~P A ) )
52, 4eqeq12d 2456 . 2  |-  ( a  =  A  ->  (
( F `  {
a } )  =  ( card `  ~P a )  <->  ( F `  { A } )  =  ( card `  ~P A ) ) )
6 ackbij1lem4 8134 . . . 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 8137 . . . 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 2965 . . . . . 6  |-  a  e. 
_V
11 sneq 3849 . . . . . . 7  |-  ( y  =  a  ->  { y }  =  { a } )
12 pweq 3826 . . . . . . 7  |-  ( y  =  a  ->  ~P y  =  ~P a
)
1311, 12xpeq12d 4932 . . . . . 6  |-  ( y  =  a  ->  ( { y }  X.  ~P y )  =  ( { a }  X.  ~P a ) )
1410, 13iunxsn 4195 . . . . 5  |-  U_ y  e.  { a }  ( { y }  X.  ~P y )  =  ( { a }  X.  ~P a )
1514fveq2i 5760 . . . 4  |-  ( card `  U_ y  e.  {
a }  ( { y }  X.  ~P y ) )  =  ( card `  ( { a }  X.  ~P a ) )
1610pwex 4411 . . . . . 6  |-  ~P a  e.  _V
17 xpsnen2g 7230 . . . . . 6  |-  ( ( a  e.  _V  /\  ~P a  e.  _V )  ->  ( { a }  X.  ~P a
)  ~~  ~P a
)
1810, 16, 17mp2an 655 . . . . 5  |-  ( { a }  X.  ~P a )  ~~  ~P a
19 carden2b 7885 . . . . 5  |-  ( ( { a }  X.  ~P a )  ~~  ~P a  ->  ( card `  ( { a }  X.  ~P a ) )  =  ( card `  ~P a ) )
2018, 19ax-mp 5 . . . 4  |-  ( card `  ( { a }  X.  ~P a ) )  =  ( card `  ~P a )
2115, 20eqtri 2462 . . 3  |-  ( card `  U_ y  e.  {
a }  ( { y }  X.  ~P y ) )  =  ( card `  ~P a )
229, 21syl6eq 2490 . 2  |-  ( a  e.  om  ->  ( F `  { a } )  =  (
card `  ~P a
) )
235, 22vtoclga 3023 1  |-  ( A  e.  om  ->  ( F `  { A } )  =  (
card `  ~P A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1727   _Vcvv 2962    i^i cin 3305   ~Pcpw 3823   {csn 3838   U_ciun 4117   class class class wbr 4237    e. cmpt 4291   omcom 4874    X. cxp 4905   ` cfv 5483    ~~ cen 7135   Fincfn 7138   cardccrd 7853
This theorem is referenced by:  ackbij1lem14  8144  ackbij1b  8150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-pss 3322  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-tp 3846  df-op 3847  df-uni 4040  df-int 4075  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-tr 4328  df-eprel 4523  df-id 4527  df-po 4532  df-so 4533  df-fr 4570  df-we 4572  df-ord 4613  df-on 4614  df-lim 4615  df-suc 4616  df-om 4875  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-1st 6378  df-2nd 6379  df-1o 6753  df-er 6934  df-en 7139  df-fin 7142  df-card 7857
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