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Metamath Proof Explorer

Theorem itcoval1

Description: A function iterated once. (Contributed by AV, 2-May-2024)

Ref Expression
Assertion itcoval1 
|- ( ( Rel F /\ F e. V ) -> ( ( IterComp ` F ) ` 1 ) = F )

Proof

Step Hyp Ref Expression
1 itcoval 
 |-  ( 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 ) ) ) )
2 1 fveq1d 
 |-  ( F e. V -> ( ( IterComp ` F ) ` 1 ) = ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` 1 ) )
3 2 adantl 
 |-  ( ( Rel F /\ F e. V ) -> ( ( IterComp ` F ) ` 1 ) = ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` 1 ) )
4 nn0uz 
 |-  NN0 = ( ZZ>= ` 0 )
5 0nn0 
 |-  0 e. NN0
6 5 a1i 
 |-  ( ( Rel F /\ F e. V ) -> 0 e. NN0 )
7 1e0p1 
 |-  1 = ( 0 + 1 )
8 1 eqcomd 
 |-  ( F e. V -> seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) = ( IterComp ` F ) )
9 8 fveq1d 
 |-  ( F e. V -> ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` 0 ) = ( ( IterComp ` F ) ` 0 ) )
10 itcoval0 
 |-  ( F e. V -> ( ( IterComp ` F ) ` 0 ) = ( _I |` dom F ) )
11 9 10 eqtrd 
 |-  ( F e. V -> ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` 0 ) = ( _I |` dom F ) )
12 11 adantl 
 |-  ( ( Rel F /\ F e. V ) -> ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` 0 ) = ( _I |` dom F ) )
13 eqidd 
 |-  ( ( Rel F /\ F e. V ) -> ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) = ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) )
14 ax-1ne0 
 |-  1 =/= 0
15 14 neii 
 |-  -. 1 = 0
16 eqeq1 
 |-  ( i = 1 -> ( i = 0 <-> 1 = 0 ) )
17 15 16 mtbiri 
 |-  ( i = 1 -> -. i = 0 )
18 17 iffalsed 
 |-  ( i = 1 -> if ( i = 0 , ( _I |` dom F ) , F ) = F )
19 18 adantl 
 |-  ( ( ( Rel F /\ F e. V ) /\ i = 1 ) -> if ( i = 0 , ( _I |` dom F ) , F ) = F )
20 1nn0 
 |-  1 e. NN0
21 20 a1i 
 |-  ( ( Rel F /\ F e. V ) -> 1 e. NN0 )
22 simpr 
 |-  ( ( Rel F /\ F e. V ) -> F e. V )
23 13 19 21 22 fvmptd 
 |-  ( ( Rel F /\ F e. V ) -> ( ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ` 1 ) = F )
24 4 6 7 12 23 seqp1d 
 |-  ( ( Rel F /\ F e. V ) -> ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` 1 ) = ( ( _I |` dom F ) ( g e. _V , j e. _V |-> ( F o. g ) ) F ) )
25 eqidd 
 |-  ( F e. V -> ( g e. _V , j e. _V |-> ( F o. g ) ) = ( g e. _V , j e. _V |-> ( F o. g ) ) )
26 coeq2 
 |-  ( g = ( _I |` dom F ) -> ( F o. g ) = ( F o. ( _I |` dom F ) ) )
27 26 ad2antrl 
 |-  ( ( F e. V /\ ( g = ( _I |` dom F ) /\ j = F ) ) -> ( F o. g ) = ( F o. ( _I |` dom F ) ) )
28 dmexg 
 |-  ( F e. V -> dom F e. _V )
29 28 resiexd 
 |-  ( F e. V -> ( _I |` dom F ) e. _V )
30 elex 
 |-  ( F e. V -> F e. _V )
31 coexg 
 |-  ( ( F e. V /\ ( _I |` dom F ) e. _V ) -> ( F o. ( _I |` dom F ) ) e. _V )
32 29 31 mpdan 
 |-  ( F e. V -> ( F o. ( _I |` dom F ) ) e. _V )
33 25 27 29 30 32 ovmpod 
 |-  ( F e. V -> ( ( _I |` dom F ) ( g e. _V , j e. _V |-> ( F o. g ) ) F ) = ( F o. ( _I |` dom F ) ) )
34 33 adantl 
 |-  ( ( Rel F /\ F e. V ) -> ( ( _I |` dom F ) ( g e. _V , j e. _V |-> ( F o. g ) ) F ) = ( F o. ( _I |` dom F ) ) )
35 coires1 
 |-  ( F o. ( _I |` dom F ) ) = ( F |` dom F )
36 resdm 
 |-  ( Rel F -> ( F |` dom F ) = F )
37 36 adantr 
 |-  ( ( Rel F /\ F e. V ) -> ( F |` dom F ) = F )
38 35 37 eqtrid 
 |-  ( ( Rel F /\ F e. V ) -> ( F o. ( _I |` dom F ) ) = F )
39 34 38 eqtrd 
 |-  ( ( Rel F /\ F e. V ) -> ( ( _I |` dom F ) ( g e. _V , j e. _V |-> ( F o. g ) ) F ) = F )
40 24 39 eqtrd 
 |-  ( ( Rel F /\ F e. V ) -> ( seq 0 ( ( g e. _V , j e. _V |-> ( F o. g ) ) , ( i e. NN0 |-> if ( i = 0 , ( _I |` dom F ) , F ) ) ) ` 1 ) = F )
41 3 40 eqtrd 
 |-  ( ( Rel F /\ F e. V ) -> ( ( IterComp ` F ) ` 1 ) = F )