Step |
Hyp |
Ref |
Expression |
1 |
|
itcovalendof.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
itcovalendof.f |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐴 ) |
3 |
|
itcovalendof.n |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
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 |
⊢ ( 𝜑 → ( I ↾ 𝐴 ) : 𝐴 ⟶ 𝐴 ) |
15 |
2
|
fdmd |
⊢ ( 𝜑 → dom 𝐹 = 𝐴 ) |
16 |
15
|
reseq2d |
⊢ ( 𝜑 → ( I ↾ dom 𝐹 ) = ( I ↾ 𝐴 ) ) |
17 |
16
|
feq1d |
⊢ ( 𝜑 → ( ( I ↾ dom 𝐹 ) : 𝐴 ⟶ 𝐴 ↔ ( I ↾ 𝐴 ) : 𝐴 ⟶ 𝐴 ) ) |
18 |
14 17
|
mpbird |
⊢ ( 𝜑 → ( I ↾ dom 𝐹 ) : 𝐴 ⟶ 𝐴 ) |
19 |
2 1
|
fexd |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
20 |
|
itcoval0 |
⊢ ( 𝐹 ∈ V → ( ( IterComp ‘ 𝐹 ) ‘ 0 ) = ( I ↾ dom 𝐹 ) ) |
21 |
19 20
|
syl |
⊢ ( 𝜑 → ( ( IterComp ‘ 𝐹 ) ‘ 0 ) = ( I ↾ dom 𝐹 ) ) |
22 |
21
|
feq1d |
⊢ ( 𝜑 → ( ( ( IterComp ‘ 𝐹 ) ‘ 0 ) : 𝐴 ⟶ 𝐴 ↔ ( I ↾ dom 𝐹 ) : 𝐴 ⟶ 𝐴 ) ) |
23 |
18 22
|
mpbird |
⊢ ( 𝜑 → ( ( IterComp ‘ 𝐹 ) ‘ 0 ) : 𝐴 ⟶ 𝐴 ) |
24 |
2
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) : 𝐴 ⟶ 𝐴 ) → 𝐹 : 𝐴 ⟶ 𝐴 ) |
25 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) : 𝐴 ⟶ 𝐴 ) → ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) : 𝐴 ⟶ 𝐴 ) |
26 |
24 25
|
fcod |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) : 𝐴 ⟶ 𝐴 ) → ( 𝐹 ∘ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) ) : 𝐴 ⟶ 𝐴 ) |
27 |
19
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) : 𝐴 ⟶ 𝐴 ) → 𝐹 ∈ V ) |
28 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) : 𝐴 ⟶ 𝐴 ) → 𝑦 ∈ ℕ0 ) |
29 |
|
eqidd |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) : 𝐴 ⟶ 𝐴 ) → ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) ) |
30 |
|
itcovalsucov |
⊢ ( ( 𝐹 ∈ V ∧ 𝑦 ∈ ℕ0 ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) = ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) ) → ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ∘ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) ) ) |
31 |
27 28 29 30
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) : 𝐴 ⟶ 𝐴 ) → ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑦 + 1 ) ) = ( 𝐹 ∘ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) ) ) |
32 |
31
|
feq1d |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) : 𝐴 ⟶ 𝐴 ) → ( ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑦 + 1 ) ) : 𝐴 ⟶ 𝐴 ↔ ( 𝐹 ∘ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) ) : 𝐴 ⟶ 𝐴 ) ) |
33 |
26 32
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℕ0 ) ∧ ( ( IterComp ‘ 𝐹 ) ‘ 𝑦 ) : 𝐴 ⟶ 𝐴 ) → ( ( IterComp ‘ 𝐹 ) ‘ ( 𝑦 + 1 ) ) : 𝐴 ⟶ 𝐴 ) |
34 |
5 7 9 11 23 33
|
nn0indd |
⊢ ( ( 𝜑 ∧ 𝑁 ∈ ℕ0 ) → ( ( IterComp ‘ 𝐹 ) ‘ 𝑁 ) : 𝐴 ⟶ 𝐴 ) |
35 |
3 34
|
mpdan |
⊢ ( 𝜑 → ( ( IterComp ‘ 𝐹 ) ‘ 𝑁 ) : 𝐴 ⟶ 𝐴 ) |