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Theorem tendo02 35939
Description: Value of additive identity endomorphism. (Contributed by NM, 11-Jun-2013.)
Hypotheses
Ref Expression
tendo0cbv.o  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
tendo02.b  |-  B  =  ( Base `  K
)
Assertion
Ref Expression
tendo02  |-  ( F  e.  T  ->  ( O `  F )  =  (  _I  |`  B ) )
Distinct variable groups:    B, f    T, f
Allowed substitution hints:    F( f)    K( f)    O( f)

Proof of Theorem tendo02
Dummy variable  g is distinct from all other variables.
StepHypRef Expression
1 eqidd 2468 . 2  |-  ( g  =  F  ->  (  _I  |`  B )  =  (  _I  |`  B ) )
2 tendo0cbv.o . . 3  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
32tendo0cbv 35938 . 2  |-  O  =  ( g  e.  T  |->  (  _I  |`  B ) )
4 funi 5624 . . 3  |-  Fun  _I
5 tendo02.b . . . 4  |-  B  =  ( Base `  K
)
6 fvex 5882 . . . 4  |-  ( Base `  K )  e.  _V
75, 6eqeltri 2551 . . 3  |-  B  e. 
_V
8 resfunexg 6137 . . 3  |-  ( ( Fun  _I  /\  B  e.  _V )  ->  (  _I  |`  B )  e. 
_V )
94, 7, 8mp2an 672 . 2  |-  (  _I  |`  B )  e.  _V
101, 3, 9fvmpt 5957 1  |-  ( F  e.  T  ->  ( O `  F )  =  (  _I  |`  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   _Vcvv 3118    |-> cmpt 4511    _I cid 4796    |` cres 5007   Fun wfun 5588   ` cfv 5594   Basecbs 14507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602
This theorem is referenced by:  tendo0co2  35940  tendo0tp  35941  tendo0pl  35943  tendoipl  35949  tendoid0  35977  tendo0mul  35978  tendo0mulr  35979  tendo1ne0  35980  tendoex  36127  dicn0  36345  dihordlem7b  36368  dihmeetlem1N  36443  dihglblem5apreN  36444  dihmeetlem4preN  36459  dihmeetlem13N  36472
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