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Theorem tendo02 34789
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 2455 . 2  |-  ( g  =  F  ->  (  _I  |`  B )  =  (  _I  |`  B ) )
2 tendo0cbv.o . . 3  |-  O  =  ( f  e.  T  |->  (  _I  |`  B ) )
32tendo0cbv 34788 . 2  |-  O  =  ( g  e.  T  |->  (  _I  |`  B ) )
4 funi 5559 . . 3  |-  Fun  _I
5 tendo02.b . . . 4  |-  B  =  ( Base `  K
)
6 fvex 5812 . . . 4  |-  ( Base `  K )  e.  _V
75, 6eqeltri 2538 . . 3  |-  B  e. 
_V
8 resfunexg 6055 . . 3  |-  ( ( Fun  _I  /\  B  e.  _V )  ->  (  _I  |`  B )  e. 
_V )
94, 7, 8mp2an 672 . 2  |-  (  _I  |`  B )  e.  _V
101, 3, 9fvmpt 5886 1  |-  ( F  e.  T  ->  ( O `  F )  =  (  _I  |`  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   _Vcvv 3078    |-> cmpt 4461    _I cid 4742    |` cres 4953   Fun wfun 5523   ` cfv 5529   Basecbs 14295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537
This theorem is referenced by:  tendo0co2  34790  tendo0tp  34791  tendo0pl  34793  tendoipl  34799  tendoid0  34827  tendo0mul  34828  tendo0mulr  34829  tendo1ne0  34830  tendoex  34977  dicn0  35195  dihordlem7b  35218  dihmeetlem1N  35293  dihglblem5apreN  35294  dihmeetlem4preN  35309  dihmeetlem13N  35322
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