Theorem tendoicbv 36081

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

Step	Hyp	Ref	Expression
1		tendoi.i	. 2 |- I = ( s e. E |-> ( f e. T |-> `' ( s ` f ) ) )
2		fveq1 6190	. . . . . 6 |- ( s = u -> ( s ` f ) = ( u ` f ) )
3	2	cnveqd 5298	. . . . 5 |- ( s = u -> `' ( s ` f ) = `' ( u ` f ) )
4	3	mpteq2dv 4745	. . . 4 |- ( s = u -> ( f e. T |-> `' ( s ` f ) ) = ( f e. T |-> `' ( u ` f ) ) )
5		fveq2 6191	. . . . . 6 |- ( f = g -> ( u ` f ) = ( u ` g ) )
6	5	cnveqd 5298	. . . . 5 |- ( f = g -> `' ( u ` f ) = `' ( u ` g ) )
7	6	cbvmptv 4750	. . . 4 |- ( f e. T |-> `' ( u ` f ) ) = ( g e. T |-> `' ( u ` g ) )
8	4, 7	syl6eq 2672	. . 3 |- ( s = u -> ( f e. T |-> `' ( s ` f ) ) = ( g e. T |-> `' ( u ` g ) ) )
9	8	cbvmptv 4750	. 2 |- ( s e. E |-> ( f e. T |-> `' ( s ` f ) ) ) = ( u e. E |-> ( g e. T |-> `' ( u ` g ) ) )
10	1, 9	eqtri 2644	1 |- I = ( u e. E |-> ( g e. T |-> `' ( u ` g ) ) )

Syntax hints:   = wceq 1483   |-> cmpt 4729   `'ccnv 5113   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-cnv 5122  df-iota 5851  df-fv 5896

This theorem is referenced by:  tendoi  36082