Step |
Hyp |
Ref |
Expression |
1 |
|
itcovalendof.a |
|- ( ph -> A e. V ) |
2 |
|
itcovalendof.f |
|- ( ph -> F : A --> A ) |
3 |
|
itcovalendof.n |
|- ( ph -> N e. NN0 ) |
4 |
|
fveq2 |
|- ( x = 0 -> ( ( IterComp ` F ) ` x ) = ( ( IterComp ` F ) ` 0 ) ) |
5 |
4
|
feq1d |
|- ( x = 0 -> ( ( ( IterComp ` F ) ` x ) : A --> A <-> ( ( IterComp ` F ) ` 0 ) : A --> A ) ) |
6 |
|
fveq2 |
|- ( x = y -> ( ( IterComp ` F ) ` x ) = ( ( IterComp ` F ) ` y ) ) |
7 |
6
|
feq1d |
|- ( x = y -> ( ( ( IterComp ` F ) ` x ) : A --> A <-> ( ( IterComp ` F ) ` y ) : A --> A ) ) |
8 |
|
fveq2 |
|- ( x = ( y + 1 ) -> ( ( IterComp ` F ) ` x ) = ( ( IterComp ` F ) ` ( y + 1 ) ) ) |
9 |
8
|
feq1d |
|- ( x = ( y + 1 ) -> ( ( ( IterComp ` F ) ` x ) : A --> A <-> ( ( IterComp ` F ) ` ( y + 1 ) ) : A --> A ) ) |
10 |
|
fveq2 |
|- ( x = N -> ( ( IterComp ` F ) ` x ) = ( ( IterComp ` F ) ` N ) ) |
11 |
10
|
feq1d |
|- ( x = N -> ( ( ( IterComp ` F ) ` x ) : A --> A <-> ( ( IterComp ` F ) ` N ) : A --> A ) ) |
12 |
|
f1oi |
|- ( _I |` A ) : A -1-1-onto-> A |
13 |
|
f1of |
|- ( ( _I |` A ) : A -1-1-onto-> A -> ( _I |` A ) : A --> A ) |
14 |
12 13
|
mp1i |
|- ( ph -> ( _I |` A ) : A --> A ) |
15 |
2
|
fdmd |
|- ( ph -> dom F = A ) |
16 |
15
|
reseq2d |
|- ( ph -> ( _I |` dom F ) = ( _I |` A ) ) |
17 |
16
|
feq1d |
|- ( ph -> ( ( _I |` dom F ) : A --> A <-> ( _I |` A ) : A --> A ) ) |
18 |
14 17
|
mpbird |
|- ( ph -> ( _I |` dom F ) : A --> A ) |
19 |
2 1
|
fexd |
|- ( ph -> F e. _V ) |
20 |
|
itcoval0 |
|- ( F e. _V -> ( ( IterComp ` F ) ` 0 ) = ( _I |` dom F ) ) |
21 |
19 20
|
syl |
|- ( ph -> ( ( IterComp ` F ) ` 0 ) = ( _I |` dom F ) ) |
22 |
21
|
feq1d |
|- ( ph -> ( ( ( IterComp ` F ) ` 0 ) : A --> A <-> ( _I |` dom F ) : A --> A ) ) |
23 |
18 22
|
mpbird |
|- ( ph -> ( ( IterComp ` F ) ` 0 ) : A --> A ) |
24 |
2
|
ad2antrr |
|- ( ( ( ph /\ y e. NN0 ) /\ ( ( IterComp ` F ) ` y ) : A --> A ) -> F : A --> A ) |
25 |
|
simpr |
|- ( ( ( ph /\ y e. NN0 ) /\ ( ( IterComp ` F ) ` y ) : A --> A ) -> ( ( IterComp ` F ) ` y ) : A --> A ) |
26 |
24 25
|
fcod |
|- ( ( ( ph /\ y e. NN0 ) /\ ( ( IterComp ` F ) ` y ) : A --> A ) -> ( F o. ( ( IterComp ` F ) ` y ) ) : A --> A ) |
27 |
19
|
ad2antrr |
|- ( ( ( ph /\ y e. NN0 ) /\ ( ( IterComp ` F ) ` y ) : A --> A ) -> F e. _V ) |
28 |
|
simplr |
|- ( ( ( ph /\ y e. NN0 ) /\ ( ( IterComp ` F ) ` y ) : A --> A ) -> y e. NN0 ) |
29 |
|
eqidd |
|- ( ( ( ph /\ y e. NN0 ) /\ ( ( IterComp ` F ) ` y ) : A --> A ) -> ( ( IterComp ` F ) ` y ) = ( ( IterComp ` F ) ` y ) ) |
30 |
|
itcovalsucov |
|- ( ( F e. _V /\ y e. NN0 /\ ( ( IterComp ` F ) ` y ) = ( ( IterComp ` F ) ` y ) ) -> ( ( IterComp ` F ) ` ( y + 1 ) ) = ( F o. ( ( IterComp ` F ) ` y ) ) ) |
31 |
27 28 29 30
|
syl3anc |
|- ( ( ( ph /\ y e. NN0 ) /\ ( ( IterComp ` F ) ` y ) : A --> A ) -> ( ( IterComp ` F ) ` ( y + 1 ) ) = ( F o. ( ( IterComp ` F ) ` y ) ) ) |
32 |
31
|
feq1d |
|- ( ( ( ph /\ y e. NN0 ) /\ ( ( IterComp ` F ) ` y ) : A --> A ) -> ( ( ( IterComp ` F ) ` ( y + 1 ) ) : A --> A <-> ( F o. ( ( IterComp ` F ) ` y ) ) : A --> A ) ) |
33 |
26 32
|
mpbird |
|- ( ( ( ph /\ y e. NN0 ) /\ ( ( IterComp ` F ) ` y ) : A --> A ) -> ( ( IterComp ` F ) ` ( y + 1 ) ) : A --> A ) |
34 |
5 7 9 11 23 33
|
nn0indd |
|- ( ( ph /\ N e. NN0 ) -> ( ( IterComp ` F ) ` N ) : A --> A ) |
35 |
3 34
|
mpdan |
|- ( ph -> ( ( IterComp ` F ) ` N ) : A --> A ) |