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Theorem tendoi 31653
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 5729 . . . 4  |-  ( u  =  S  ->  (
u `  g )  =  ( S `  g ) )
21cnveqd 5050 . . 3  |-  ( u  =  S  ->  `' ( u `  g
)  =  `' ( S `  g ) )
32mpteq2dv 4298 . 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 31652 . 2  |-  I  =  ( u  e.  E  |->  ( g  e.  T  |->  `' ( u `  g ) ) )
6 tendoi.t . . . 4  |-  T  =  ( ( LTrn `  K
) `  W )
7 fvex 5744 . . . 4  |-  ( (
LTrn `  K ) `  W )  e.  _V
86, 7eqeltri 2508 . . 3  |-  T  e. 
_V
98mptex 5968 . 2  |-  ( g  e.  T  |->  `' ( S `  g ) )  e.  _V
103, 5, 9fvmpt 5808 1  |-  ( S  e.  E  ->  (
I `  S )  =  ( g  e.  T  |->  `' ( S `
 g ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   _Vcvv 2958    e. cmpt 4268   `'ccnv 4879   ` cfv 5456   LTrncltrn 30960
This theorem is referenced by:  tendoi2  31654  tendoicl  31655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464
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