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Theorem fin1a2lem3 8283
Description: Lemma for fin1a2 8296. (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 6090 . 2  |-  ( a  =  A  ->  ( 2o  .o  a )  =  ( 2o  .o  A
) )
2 fin1a2lem.b . . 3  |-  E  =  ( x  e.  om  |->  ( 2o  .o  x
) )
3 oveq2 6090 . . . 4  |-  ( x  =  a  ->  ( 2o  .o  x )  =  ( 2o  .o  a
) )
43cbvmptv 4301 . . 3  |-  ( x  e.  om  |->  ( 2o 
.o  x ) )  =  ( a  e. 
om  |->  ( 2o  .o  a ) )
52, 4eqtri 2457 . 2  |-  E  =  ( a  e.  om  |->  ( 2o  .o  a
) )
6 ovex 6107 . 2  |-  ( 2o 
.o  A )  e. 
_V
71, 5, 6fvmpt 5807 1  |-  ( A  e.  om  ->  ( E `  A )  =  ( 2o  .o  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726    e. cmpt 4267   omcom 4846   ` cfv 5455  (class class class)co 6082   2oc2o 6719    .o comu 6723
This theorem is referenced by:  fin1a2lem4  8284  fin1a2lem5  8285  fin1a2lem6  8286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-iota 5419  df-fun 5457  df-fv 5463  df-ov 6085
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