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Theorem ackbij1lem7 7852
Description: Lemma for ackbij1 7864. (Contributed by Stefan O'Rear, 21-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
ackbij1lem7  |-  ( A  e.  ( ~P om  i^i  Fin )  ->  ( F `  A )  =  ( card `  U_ y  e.  A  ( {
y }  X.  ~P y ) ) )
Distinct variable groups:    x, F, y    x, A, y

Proof of Theorem ackbij1lem7
StepHypRef Expression
1 iuneq1 3918 . . 3  |-  ( x  =  A  ->  U_ y  e.  x  ( {
y }  X.  ~P y )  =  U_ y  e.  A  ( { y }  X.  ~P y ) )
21fveq2d 5529 . 2  |-  ( x  =  A  ->  ( card `  U_ y  e.  x  ( { y }  X.  ~P y
) )  =  (
card `  U_ y  e.  A  ( { y }  X.  ~P y
) ) )
3 ackbij.f . 2  |-  F  =  ( x  e.  ( ~P om  i^i  Fin )  |->  ( card `  U_ y  e.  x  ( {
y }  X.  ~P y ) ) )
4 fvex 5539 . 2  |-  ( card `  U_ y  e.  A  ( { y }  X.  ~P y ) )  e. 
_V
52, 3, 4fvmpt 5602 1  |-  ( A  e.  ( ~P om  i^i  Fin )  ->  ( F `  A )  =  ( card `  U_ y  e.  A  ( {
y }  X.  ~P y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    i^i cin 3151   ~Pcpw 3625   {csn 3640   U_ciun 3905    e. cmpt 4077   omcom 4656    X. cxp 4687   ` cfv 5255   Fincfn 6863   cardccrd 7568
This theorem is referenced by:  ackbij1lem8  7853  ackbij1lem9  7854
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263
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