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Theorem fin1a2lem3 8773
Description: Lemma for fin1a2 8786. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypothesis
Ref Expression
fin1a2lem.b  |-  E  =  ( x  e.  om  |->  ( 2o  .o  x
) )
Assertion
Ref Expression
fin1a2lem3  |-  ( A  e.  om  ->  ( E `  A )  =  ( 2o  .o  A ) )

Proof of Theorem fin1a2lem3
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 oveq2 6278 . 2  |-  ( a  =  A  ->  ( 2o  .o  a )  =  ( 2o  .o  A
) )
2 fin1a2lem.b . . 3  |-  E  =  ( x  e.  om  |->  ( 2o  .o  x
) )
3 oveq2 6278 . . . 4  |-  ( x  =  a  ->  ( 2o  .o  x )  =  ( 2o  .o  a
) )
43cbvmptv 4530 . . 3  |-  ( x  e.  om  |->  ( 2o 
.o  x ) )  =  ( a  e. 
om  |->  ( 2o  .o  a ) )
52, 4eqtri 2483 . 2  |-  E  =  ( a  e.  om  |->  ( 2o  .o  a
) )
6 ovex 6298 . 2  |-  ( 2o 
.o  A )  e. 
_V
71, 5, 6fvmpt 5931 1  |-  ( A  e.  om  ->  ( E `  A )  =  ( 2o  .o  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823    |-> cmpt 4497   ` cfv 5570  (class class class)co 6270   omcom 6673   2oc2o 7116    .o comu 7120
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-ov 6273
This theorem is referenced by:  fin1a2lem4  8774  fin1a2lem5  8775  fin1a2lem6  8776
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