Step |
Hyp |
Ref |
Expression |
1 |
|
ofcfval4.1 |
⊢ ( 𝜑 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
2 |
|
ofcfval4.2 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
3 |
|
ofcfval4.3 |
⊢ ( 𝜑 → 𝐶 ∈ 𝑊 ) |
4 |
1
|
fdmd |
⊢ ( 𝜑 → dom 𝐹 = 𝐴 ) |
5 |
4
|
mpteq1d |
⊢ ( 𝜑 → ( 𝑦 ∈ dom 𝐹 ↦ ( ( 𝐹 ‘ 𝑦 ) 𝑅 𝐶 ) ) = ( 𝑦 ∈ 𝐴 ↦ ( ( 𝐹 ‘ 𝑦 ) 𝑅 𝐶 ) ) ) |
6 |
1 2
|
fexd |
⊢ ( 𝜑 → 𝐹 ∈ V ) |
7 |
|
ofcfval3 |
⊢ ( ( 𝐹 ∈ V ∧ 𝐶 ∈ 𝑊 ) → ( 𝐹 ∘f/c 𝑅 𝐶 ) = ( 𝑦 ∈ dom 𝐹 ↦ ( ( 𝐹 ‘ 𝑦 ) 𝑅 𝐶 ) ) ) |
8 |
6 3 7
|
syl2anc |
⊢ ( 𝜑 → ( 𝐹 ∘f/c 𝑅 𝐶 ) = ( 𝑦 ∈ dom 𝐹 ↦ ( ( 𝐹 ‘ 𝑦 ) 𝑅 𝐶 ) ) ) |
9 |
1
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑦 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝐵 ) |
10 |
1
|
feqmptd |
⊢ ( 𝜑 → 𝐹 = ( 𝑦 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑦 ) ) ) |
11 |
|
eqidd |
⊢ ( 𝜑 → ( 𝑥 ∈ 𝐵 ↦ ( 𝑥 𝑅 𝐶 ) ) = ( 𝑥 ∈ 𝐵 ↦ ( 𝑥 𝑅 𝐶 ) ) ) |
12 |
|
oveq1 |
⊢ ( 𝑥 = ( 𝐹 ‘ 𝑦 ) → ( 𝑥 𝑅 𝐶 ) = ( ( 𝐹 ‘ 𝑦 ) 𝑅 𝐶 ) ) |
13 |
9 10 11 12
|
fmptco |
⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑥 𝑅 𝐶 ) ) ∘ 𝐹 ) = ( 𝑦 ∈ 𝐴 ↦ ( ( 𝐹 ‘ 𝑦 ) 𝑅 𝐶 ) ) ) |
14 |
5 8 13
|
3eqtr4d |
⊢ ( 𝜑 → ( 𝐹 ∘f/c 𝑅 𝐶 ) = ( ( 𝑥 ∈ 𝐵 ↦ ( 𝑥 𝑅 𝐶 ) ) ∘ 𝐹 ) ) |