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Theorem tendoicbv 30982
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 5524 . . . . . 6  |-  ( s  =  u  ->  (
s `  f )  =  ( u `  f ) )
32cnveqd 4857 . . . . 5  |-  ( s  =  u  ->  `' ( s `  f
)  =  `' ( u `  f ) )
43mpteq2dv 4107 . . . 4  |-  ( s  =  u  ->  (
f  e.  T  |->  `' ( s `  f
) )  =  ( f  e.  T  |->  `' ( u `  f
) ) )
5 fveq2 5525 . . . . . 6  |-  ( f  =  g  ->  (
u `  f )  =  ( u `  g ) )
65cnveqd 4857 . . . . 5  |-  ( f  =  g  ->  `' ( u `  f
)  =  `' ( u `  g ) )
76cbvmptv 4111 . . . 4  |-  ( f  e.  T  |->  `' ( u `  f ) )  =  ( g  e.  T  |->  `' ( u `  g ) )
84, 7syl6eq 2331 . . 3  |-  ( s  =  u  ->  (
f  e.  T  |->  `' ( s `  f
) )  =  ( g  e.  T  |->  `' ( u `  g
) ) )
98cbvmptv 4111 . 2  |-  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f ) ) )  =  ( u  e.  E  |->  ( g  e.  T  |->  `' ( u `  g
) ) )
101, 9eqtri 2303 1  |-  I  =  ( u  e.  E  |->  ( g  e.  T  |->  `' ( u `  g ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1623    e. cmpt 4077   `'ccnv 4688   ` cfv 5255
This theorem is referenced by:  tendoi  30983
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  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-br 4024  df-opab 4078  df-mpt 4079  df-cnv 4697  df-iota 5219  df-fv 5263
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