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Theorem tendoi 30983
Description: Value of inverse endomorphism. (Contributed by NM, 12-Jun-2013.)
Hypotheses
Ref Expression
tendoi.i  |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f ) ) )
tendoi.t  |-  T  =  ( ( LTrn `  K
) `  W )
Assertion
Ref Expression
tendoi  |-  ( S  e.  E  ->  (
I `  S )  =  ( g  e.  T  |->  `' ( S `
 g ) ) )
Distinct variable groups:    E, s    f, g, s, T    f, W, g, s    S, g
Allowed substitution hints:    S( f, s)    E( f, g)    I( f, g, s)    K( f, g, s)

Proof of Theorem tendoi
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 fveq1 5524 . . . 4  |-  ( u  =  S  ->  (
u `  g )  =  ( S `  g ) )
21cnveqd 4857 . . 3  |-  ( u  =  S  ->  `' ( u `  g
)  =  `' ( S `  g ) )
32mpteq2dv 4107 . 2  |-  ( u  =  S  ->  (
g  e.  T  |->  `' ( u `  g
) )  =  ( g  e.  T  |->  `' ( S `  g
) ) )
4 tendoi.i . . 3  |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f ) ) )
54tendoicbv 30982 . 2  |-  I  =  ( u  e.  E  |->  ( g  e.  T  |->  `' ( u `  g ) ) )
6 tendoi.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
7 fvex 5539 . . . 4  |-  ( (
LTrn `  K ) `  W )  e.  _V
86, 7eqeltri 2353 . . 3  |-  T  e. 
_V
98mptex 5746 . 2  |-  ( g  e.  T  |->  `' ( S `  g ) )  e.  _V
103, 5, 9fvmpt 5602 1  |-  ( S  e.  E  ->  (
I `  S )  =  ( g  e.  T  |->  `' ( S `
 g ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    e. cmpt 4077   `'ccnv 4688   ` cfv 5255   LTrncltrn 30290
This theorem is referenced by:  tendoi2  30984  tendoicl  30985
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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