Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  tendoicbv Unicode version

Theorem tendoicbv 31604
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 5540 . . . . . 6  |-  ( s  =  u  ->  (
s `  f )  =  ( u `  f ) )
32cnveqd 4873 . . . . 5  |-  ( s  =  u  ->  `' ( s `  f
)  =  `' ( u `  f ) )
43mpteq2dv 4123 . . . 4  |-  ( s  =  u  ->  (
f  e.  T  |->  `' ( s `  f
) )  =  ( f  e.  T  |->  `' ( u `  f
) ) )
5 fveq2 5541 . . . . . 6  |-  ( f  =  g  ->  (
u `  f )  =  ( u `  g ) )
65cnveqd 4873 . . . . 5  |-  ( f  =  g  ->  `' ( u `  f
)  =  `' ( u `  g ) )
76cbvmptv 4127 . . . 4  |-  ( f  e.  T  |->  `' ( u `  f ) )  =  ( g  e.  T  |->  `' ( u `  g ) )
84, 7syl6eq 2344 . . 3  |-  ( s  =  u  ->  (
f  e.  T  |->  `' ( s `  f
) )  =  ( g  e.  T  |->  `' ( u `  g
) ) )
98cbvmptv 4127 . 2  |-  ( s  e.  E  |->  ( f  e.  T  |->  `' ( s `  f ) ) )  =  ( u  e.  E  |->  ( g  e.  T  |->  `' ( u `  g
) ) )
101, 9eqtri 2316 1  |-  I  =  ( u  e.  E  |->  ( g  e.  T  |->  `' ( u `  g ) ) )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    e. cmpt 4093   `'ccnv 4704   ` cfv 5271
This theorem is referenced by:  tendoi  31605
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-cnv 4713  df-iota 5235  df-fv 5279
  Copyright terms: Public domain W3C validator