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Theorem ackbij1lem7 8106
Description: Lemma for ackbij1 8118. (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 4106 . . 3  |-  ( x  =  A  ->  U_ y  e.  x  ( {
y }  X.  ~P y )  =  U_ y  e.  A  ( { y }  X.  ~P y ) )
21fveq2d 5732 . 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 5742 . 2  |-  ( card `  U_ y  e.  A  ( { y }  X.  ~P y ) )  e. 
_V
52, 3, 4fvmpt 5806 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 1652    e. wcel 1725    i^i cin 3319   ~Pcpw 3799   {csn 3814   U_ciun 4093    e. cmpt 4266   omcom 4845    X. cxp 4876   ` cfv 5454   Fincfn 7109   cardccrd 7822
This theorem is referenced by:  ackbij1lem8  8107  ackbij1lem9  8108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462
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