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Theorem tendoicbv 30907
Description: Define inverse function for trace-perserving endomorphisms. Change bound variable to isolate it later. (Contributed by NM, 12-Jun-2013.)
Hypothesis
Ref Expression
tendoi.i  |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f ) ) )
Assertion
Ref Expression
tendoicbv  |-  I  =  ( u  e.  E  |->  ( g  e.  T  |->  `' ( u `  g ) ) )
Distinct variable groups:    u, s, E    f, g, s, u, T
Allowed substitution hints:    E( f, g)    I( u, f, g, s)

Proof of Theorem tendoicbv
StepHypRef Expression
1 tendoi.i . 2  |-  I  =  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f ) ) )
2 fveq1 5667 . . . . . 6  |-  ( s  =  u  ->  (
s `  f )  =  ( u `  f ) )
32cnveqd 4988 . . . . 5  |-  ( s  =  u  ->  `' ( s `  f
)  =  `' ( u `  f ) )
43mpteq2dv 4237 . . . 4  |-  ( s  =  u  ->  (
f  e.  T  |->  `' ( s `  f
) )  =  ( f  e.  T  |->  `' ( u `  f
) ) )
5 fveq2 5668 . . . . . 6  |-  ( f  =  g  ->  (
u `  f )  =  ( u `  g ) )
65cnveqd 4988 . . . . 5  |-  ( f  =  g  ->  `' ( u `  f
)  =  `' ( u `  g ) )
76cbvmptv 4241 . . . 4  |-  ( f  e.  T  |->  `' ( u `  f ) )  =  ( g  e.  T  |->  `' ( u `  g ) )
84, 7syl6eq 2435 . . 3  |-  ( s  =  u  ->  (
f  e.  T  |->  `' ( s `  f
) )  =  ( g  e.  T  |->  `' ( u `  g
) ) )
98cbvmptv 4241 . 2  |-  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f ) ) )  =  ( u  e.  E  |->  ( g  e.  T  |->  `' ( u `  g
) ) )
101, 9eqtri 2407 1  |-  I  =  ( u  e.  E  |->  ( g  e.  T  |->  `' ( u `  g ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. cmpt 4207   `'ccnv 4817   ` cfv 5394
This theorem is referenced by:  tendoi  30908
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-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
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-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-cnv 4826  df-iota 5358  df-fv 5402
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