Theorem tendo02 36075

Description: Value of additive identity endomorphism. (Contributed by NM, 11-Jun-2013.)

Hypotheses

Ref	Expression
tendo0cbv.o	|- O = ( f e. T |-> ( _I |` B ) )
tendo02.b	|- B = ( Base ` K )

Assertion

Ref	Expression
tendo02	|- ( F e. T -> ( O ` F ) = ( _I |` B ) )

Distinct variable groups:   B, f   T, f

Allowed substitution hints:   F( f)   K( f)   O( f)

Proof of Theorem tendo02

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2623	. 2 |- ( g = F -> ( _I |` B ) = ( _I |` B ) )
2		tendo0cbv.o	. . 3 |- O = ( f e. T |-> ( _I |` B ) )
3	2	tendo0cbv 36074	. 2 |- O = ( g e. T |-> ( _I |` B ) )
4		funi 5920	. . 3 |- Fun _I
5		tendo02.b	. . . 4 |- B = ( Base ` K )
6		fvex 6201	. . . 4 |- ( Base ` K ) e. _V
7	5, 6	eqeltri 2697	. . 3 |- B e. _V
8		resfunexg 6479	. . 3 |- ( ( Fun _I /\ B e. _V ) -> ( _I |` B ) e. _V )
9	4, 7, 8	mp2an 708	. 2 |- ( _I |` B ) e. _V
10	1, 3, 9	fvmpt 6282	1 |- ( F e. T -> ( O ` F ) = ( _I |` B ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   _I cid 5023   |` cres 5116   Fun wfun 5882   ` cfv 5888   Basecbs 15857

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  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

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-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896

This theorem is referenced by:  tendo0co2  36076  tendo0tp  36077  tendo0pl  36079  tendoipl  36085  tendoid0  36113  tendo0mul  36114  tendo0mulr  36115  tendo1ne0  36116  tendoex  36263  dicn0  36481  dihordlem7b  36504  dihmeetlem1N  36579  dihglblem5apreN  36580  dihmeetlem4preN  36595  dihmeetlem13N  36608