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Theorem ackbij1lem8 8000
Description: Lemma for ackbij1 8011. (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 3740 . . . 4  |-  ( a  =  A  ->  { a }  =  { A } )
21fveq2d 5636 . . 3  |-  ( a  =  A  ->  ( F `  { a } )  =  ( F `  { A } ) )
3 pweq 3717 . . . 4  |-  ( a  =  A  ->  ~P a  =  ~P A
)
43fveq2d 5636 . . 3  |-  ( a  =  A  ->  ( card `  ~P a )  =  ( card `  ~P A ) )
52, 4eqeq12d 2380 . 2  |-  ( a  =  A  ->  (
( F `  {
a } )  =  ( card `  ~P a )  <->  ( F `  { A } )  =  ( card `  ~P A ) ) )
6 ackbij1lem4 7996 . . . 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 7999 . . . 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 2876 . . . . . 6  |-  a  e. 
_V
11 sneq 3740 . . . . . . 7  |-  ( y  =  a  ->  { y }  =  { a } )
12 pweq 3717 . . . . . . 7  |-  ( y  =  a  ->  ~P y  =  ~P a
)
1311, 12xpeq12d 4817 . . . . . 6  |-  ( y  =  a  ->  ( { y }  X.  ~P y )  =  ( { a }  X.  ~P a ) )
1410, 13iunxsn 4083 . . . . 5  |-  U_ y  e.  { a }  ( { y }  X.  ~P y )  =  ( { a }  X.  ~P a )
1514fveq2i 5635 . . . 4  |-  ( card `  U_ y  e.  {
a }  ( { y }  X.  ~P y ) )  =  ( card `  ( { a }  X.  ~P a ) )
1610pwex 4295 . . . . . 6  |-  ~P a  e.  _V
17 xpsnen2g 7098 . . . . . 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 7747 . . . . 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 2386 . . 3  |-  ( card `  U_ y  e.  {
a }  ( { y }  X.  ~P y ) )  =  ( card `  ~P a )
229, 21syl6eq 2414 . 2  |-  ( a  e.  om  ->  ( F `  { a } )  =  (
card `  ~P a
) )
235, 22vtoclga 2934 1  |-  ( A  e.  om  ->  ( F `  { A } )  =  (
card `  ~P A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. wcel 1715   _Vcvv 2873    i^i cin 3237   ~Pcpw 3714   {csn 3729   U_ciun 4007   class class class wbr 4125    e. cmpt 4179   omcom 4759    X. cxp 4790   ` cfv 5358    ~~ cen 7003   Fincfn 7006   cardccrd 7715
This theorem is referenced by:  ackbij1lem14  8006  ackbij1b  8012
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-tp 3737  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-tr 4216  df-eprel 4408  df-id 4412  df-po 4417  df-so 4418  df-fr 4455  df-we 4457  df-ord 4498  df-on 4499  df-lim 4500  df-suc 4501  df-om 4760  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-1st 6249  df-2nd 6250  df-1o 6621  df-er 6802  df-en 7007  df-fin 7010  df-card 7719
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