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Theorem tendo02 31598
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 2297 . 2  |-  ( g  =  F  ->  (  _I  |`  B )  =  (  _I  |`  B ) )
2 tendo0cbv.o . . 3  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
32tendo0cbv 31597 . 2  |-  O  =  ( g  e.  T  |->  (  _I  |`  B ) )
4 funi 5300 . . 3  |-  Fun  _I
5 tendo02.b . . . 4  |-  B  =  ( Base `  K
)
6 fvex 5555 . . . 4  |-  ( Base `  K )  e.  _V
75, 6eqeltri 2366 . . 3  |-  B  e. 
_V
8 resfunexg 5753 . . 3  |-  ( ( Fun  _I  /\  B  e.  _V )  ->  (  _I  |`  B )  e. 
_V )
94, 7, 8mp2an 653 . 2  |-  (  _I  |`  B )  e.  _V
101, 3, 9fvmpt 5618 1  |-  ( F  e.  T  ->  ( O `  F )  =  (  _I  |`  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801    e. cmpt 4093    _I cid 4320    |` cres 4707   Fun wfun 5265   ` cfv 5271   Basecbs 13164
This theorem is referenced by:  tendo0co2  31599  tendo0tp  31600  tendo0pl  31602  tendoipl  31608  tendoid0  31636  tendo0mul  31637  tendo0mulr  31638  tendo1ne0  31639  tendoex  31786  dicn0  32004  dihordlem7b  32027  dihmeetlem1N  32102  dihglblem5apreN  32103  dihmeetlem4preN  32118  dihmeetlem13N  32131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279
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