Step |
Hyp |
Ref |
Expression |
1 |
|
offvalfv.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
offvalfv.f |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
3 |
|
offvalfv.g |
⊢ ( 𝜑 → 𝐺 Fn 𝐴 ) |
4 |
|
fnfvelrn |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ ran 𝐹 ) |
5 |
2 4
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ ran 𝐹 ) |
6 |
|
fnfvelrn |
⊢ ( ( 𝐺 Fn 𝐴 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) ∈ ran 𝐺 ) |
7 |
3 6
|
sylan |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) ∈ ran 𝐺 ) |
8 |
|
dffn5 |
⊢ ( 𝐹 Fn 𝐴 ↔ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
9 |
2 8
|
sylib |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ 𝐴 ↦ ( 𝐹 ‘ 𝑥 ) ) ) |
10 |
|
dffn5 |
⊢ ( 𝐺 Fn 𝐴 ↔ 𝐺 = ( 𝑥 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑥 ) ) ) |
11 |
3 10
|
sylib |
⊢ ( 𝜑 → 𝐺 = ( 𝑥 ∈ 𝐴 ↦ ( 𝐺 ‘ 𝑥 ) ) ) |
12 |
1 5 7 9 11
|
offval2 |
⊢ ( 𝜑 → ( 𝐹 ∘f 𝑅 𝐺 ) = ( 𝑥 ∈ 𝐴 ↦ ( ( 𝐹 ‘ 𝑥 ) 𝑅 ( 𝐺 ‘ 𝑥 ) ) ) ) |