Step |
Hyp |
Ref |
Expression |
1 |
|
df-itco |
|- IterComp = ( f e. _V |-> seq 0 ( ( g e. _V , j e. _V |-> ( f o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom f ) , f ) ) ) ) |
2 |
|
eqidd |
|- ( f = F -> 0 = 0 ) |
3 |
|
coeq1 |
|- ( f = F -> ( f o. g ) = ( F o. g ) ) |
4 |
3
|
mpoeq3dv |
|- ( f = F -> ( g e. _V , j e. _V |-> ( f o. g ) ) = ( g e. _V , j e. _V |-> ( F o. g ) ) ) |
5 |
|
dmeq |
|- ( f = F -> dom f = dom F ) |
6 |
5
|
reseq2d |
|- ( f = F -> ( _I |` dom f ) = ( _I |` dom F ) ) |
7 |
|
id |
|- ( f = F -> f = F ) |
8 |
6 7
|
ifeq12d |
|- ( f = F -> if ( i = 0 , ( _I |` dom f ) , f ) = if ( i = 0 , ( _I |` dom F ) , F ) ) |
9 |
8
|
mpteq2dv |
|- ( f = F -> ( i e. NN0 |-> if ( i = 0 , ( _I |` dom f ) , f ) ) = ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) |
10 |
2 4 9
|
seqeq123d |
|- ( f = F -> seq 0 ( ( g e. _V , j e. _V |-> ( f o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom f ) , f ) ) ) = seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ) |
11 |
|
elex |
|- ( F e. V -> F e. _V ) |
12 |
|
seqex |
|- seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) e. _V |
13 |
12
|
a1i |
|- ( F e. V -> seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) e. _V ) |
14 |
1 10 11 13
|
fvmptd3 |
|- ( F e. V -> ( IterComp ` F ) = seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ) |