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Theorem tendo02 30976
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 2284 . 2  |-  ( g  =  F  ->  (  _I  |`  B )  =  (  _I  |`  B ) )
2 tendo0cbv.o . . 3  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
32tendo0cbv 30975 . 2  |-  O  =  ( g  e.  T  |->  (  _I  |`  B ) )
4 funi 5284 . . 3  |-  Fun  _I
5 tendo02.b . . . 4  |-  B  =  ( Base `  K
)
6 fvex 5539 . . . 4  |-  ( Base `  K )  e.  _V
75, 6eqeltri 2353 . . 3  |-  B  e. 
_V
8 resfunexg 5737 . . 3  |-  ( ( Fun  _I  /\  B  e.  _V )  ->  (  _I  |`  B )  e. 
_V )
94, 7, 8mp2an 653 . 2  |-  (  _I  |`  B )  e.  _V
101, 3, 9fvmpt 5602 1  |-  ( F  e.  T  ->  ( O `  F )  =  (  _I  |`  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077    _I cid 4304    |` cres 4691   Fun wfun 5249   ` cfv 5255   Basecbs 13148
This theorem is referenced by:  tendo0co2  30977  tendo0tp  30978  tendo0pl  30980  tendoipl  30986  tendoid0  31014  tendo0mul  31015  tendo0mulr  31016  tendo1ne0  31017  tendoex  31164  dicn0  31382  dihordlem7b  31405  dihmeetlem1N  31480  dihglblem5apreN  31481  dihmeetlem4preN  31496  dihmeetlem13N  31509
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-rep 4131  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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  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-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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