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Theorem tendo02 30903
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 2390 . 2  |-  ( g  =  F  ->  (  _I  |`  B )  =  (  _I  |`  B ) )
2 tendo0cbv.o . . 3  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
32tendo0cbv 30902 . 2  |-  O  =  ( g  e.  T  |->  (  _I  |`  B ) )
4 funi 5425 . . 3  |-  Fun  _I
5 tendo02.b . . . 4  |-  B  =  ( Base `  K
)
6 fvex 5684 . . . 4  |-  ( Base `  K )  e.  _V
75, 6eqeltri 2459 . . 3  |-  B  e. 
_V
8 resfunexg 5898 . . 3  |-  ( ( Fun  _I  /\  B  e.  _V )  ->  (  _I  |`  B )  e. 
_V )
94, 7, 8mp2an 654 . 2  |-  (  _I  |`  B )  e.  _V
101, 3, 9fvmpt 5747 1  |-  ( F  e.  T  ->  ( O `  F )  =  (  _I  |`  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2901    e. cmpt 4209    _I cid 4436    |` cres 4822   Fun wfun 5390   ` cfv 5396   Basecbs 13398
This theorem is referenced by:  tendo0co2  30904  tendo0tp  30905  tendo0pl  30907  tendoipl  30913  tendoid0  30941  tendo0mul  30942  tendo0mulr  30943  tendo1ne0  30944  tendoex  31091  dicn0  31309  dihordlem7b  31332  dihmeetlem1N  31407  dihglblem5apreN  31408  dihmeetlem4preN  31423  dihmeetlem13N  31436
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404
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