Metamath Proof Explorer

Theorem itcoval1

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

Ref Expression
Assertion itcoval1 ( ( Rel 𝐹 ∧ 𝐹 ∈ 𝑉 ) → ( ( IterComp ‘ 𝐹 ) ‘ 1 ) = 𝐹 )

Proof

Step Hyp Ref Expression
1 itcoval ⊢ ( 𝐹 ∈ 𝑉 → ( IterComp ‘ 𝐹 ) = seq 0 ( ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) )
2 1 fveq1d ⊢ ( 𝐹 ∈ 𝑉 → ( ( IterComp ‘ 𝐹 ) ‘ 1 ) = ( seq 0 ( ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) ‘ 1 ) )
3 2 adantl ⊢ ( ( Rel 𝐹 ∧ 𝐹 ∈ 𝑉 ) → ( ( IterComp ‘ 𝐹 ) ‘ 1 ) = ( seq 0 ( ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) ‘ 1 ) )
4 nn0uz ⊢ ℕ0 = ( ℤ≥ ‘ 0 )
5 0nn0 ⊢ 0 ∈ ℕ0
6 5 a1i ⊢ ( ( Rel 𝐹 ∧ 𝐹 ∈ 𝑉 ) → 0 ∈ ℕ0 )
7 1e0p1 ⊢ 1 = ( 0 + 1 )
8 1 eqcomd ⊢ ( 𝐹 ∈ 𝑉 → seq 0 ( ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) = ( IterComp ‘ 𝐹 ) )
9 8 fveq1d ⊢ ( 𝐹 ∈ 𝑉 → ( seq 0 ( ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) ‘ 0 ) = ( ( IterComp ‘ 𝐹 ) ‘ 0 ) )
10 itcoval0 ⊢ ( 𝐹 ∈ 𝑉 → ( ( IterComp ‘ 𝐹 ) ‘ 0 ) = ( I ↾ dom 𝐹 ) )
11 9 10 eqtrd ⊢ ( 𝐹 ∈ 𝑉 → ( seq 0 ( ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) ‘ 0 ) = ( I ↾ dom 𝐹 ) )
12 11 adantl ⊢ ( ( Rel 𝐹 ∧ 𝐹 ∈ 𝑉 ) → ( seq 0 ( ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) ‘ 0 ) = ( I ↾ dom 𝐹 ) )
13 eqidd ⊢ ( ( Rel 𝐹 ∧ 𝐹 ∈ 𝑉 ) → ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) = ( 𝑖 ∈ ℕ0 ↦ 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 ∈ 𝑉 ) ∧ 𝑖 = 1 ) → if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) = 𝐹 )
20 1nn0 ⊢ 1 ∈ ℕ0
21 20 a1i ⊢ ( ( Rel 𝐹 ∧ 𝐹 ∈ 𝑉 ) → 1 ∈ ℕ0 )
22 simpr ⊢ ( ( Rel 𝐹 ∧ 𝐹 ∈ 𝑉 ) → 𝐹 ∈ 𝑉 )
23 13 19 21 22 fvmptd ⊢ ( ( Rel 𝐹 ∧ 𝐹 ∈ 𝑉 ) → ( ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ‘ 1 ) = 𝐹 )
24 4 6 7 12 23 seqp1d ⊢ ( ( Rel 𝐹 ∧ 𝐹 ∈ 𝑉 ) → ( seq 0 ( ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) ‘ 1 ) = ( ( I ↾ dom 𝐹 ) ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) 𝐹 ) )
25 eqidd ⊢ ( 𝐹 ∈ 𝑉 → ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) = ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) )
26 coeq2 ⊢ ( 𝑔 = ( I ↾ dom 𝐹 ) → ( 𝐹 ∘ 𝑔 ) = ( 𝐹 ∘ ( I ↾ dom 𝐹 ) ) )
27 26 ad2antrl ⊢ ( ( 𝐹 ∈ 𝑉 ∧ ( 𝑔 = ( I ↾ dom 𝐹 ) ∧ 𝑗 = 𝐹 ) ) → ( 𝐹 ∘ 𝑔 ) = ( 𝐹 ∘ ( I ↾ dom 𝐹 ) ) )
28 dmexg ⊢ ( 𝐹 ∈ 𝑉 → dom 𝐹 ∈ V )
29 28 resiexd ⊢ ( 𝐹 ∈ 𝑉 → ( I ↾ dom 𝐹 ) ∈ V )
30 elex ⊢ ( 𝐹 ∈ 𝑉 → 𝐹 ∈ V )
31 coexg ⊢ ( ( 𝐹 ∈ 𝑉 ∧ ( I ↾ dom 𝐹 ) ∈ V ) → ( 𝐹 ∘ ( I ↾ dom 𝐹 ) ) ∈ V )
32 29 31 mpdan ⊢ ( 𝐹 ∈ 𝑉 → ( 𝐹 ∘ ( I ↾ dom 𝐹 ) ) ∈ V )
33 25 27 29 30 32 ovmpod ⊢ ( 𝐹 ∈ 𝑉 → ( ( I ↾ dom 𝐹 ) ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) 𝐹 ) = ( 𝐹 ∘ ( I ↾ dom 𝐹 ) ) )
34 33 adantl ⊢ ( ( Rel 𝐹 ∧ 𝐹 ∈ 𝑉 ) → ( ( I ↾ dom 𝐹 ) ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) 𝐹 ) = ( 𝐹 ∘ ( I ↾ dom 𝐹 ) ) )
35 coires1 ⊢ ( 𝐹 ∘ ( I ↾ dom 𝐹 ) ) = ( 𝐹 ↾ dom 𝐹 )
36 resdm ⊢ ( Rel 𝐹 → ( 𝐹 ↾ dom 𝐹 ) = 𝐹 )
37 36 adantr ⊢ ( ( Rel 𝐹 ∧ 𝐹 ∈ 𝑉 ) → ( 𝐹 ↾ dom 𝐹 ) = 𝐹 )
38 35 37 eqtrid ⊢ ( ( Rel 𝐹 ∧ 𝐹 ∈ 𝑉 ) → ( 𝐹 ∘ ( I ↾ dom 𝐹 ) ) = 𝐹 )
39 34 38 eqtrd ⊢ ( ( Rel 𝐹 ∧ 𝐹 ∈ 𝑉 ) → ( ( I ↾ dom 𝐹 ) ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) 𝐹 ) = 𝐹 )
40 24 39 eqtrd ⊢ ( ( Rel 𝐹 ∧ 𝐹 ∈ 𝑉 ) → ( seq 0 ( ( 𝑔 ∈ V , 𝑗 ∈ V ↦ ( 𝐹 ∘ 𝑔 ) ) , ( 𝑖 ∈ ℕ0 ↦ if ( 𝑖 = 0 , ( I ↾ dom 𝐹 ) , 𝐹 ) ) ) ‘ 1 ) = 𝐹 )
41 3 40 eqtrd ⊢ ( ( Rel 𝐹 ∧ 𝐹 ∈ 𝑉 ) → ( ( IterComp ‘ 𝐹 ) ‘ 1 ) = 𝐹 )