Step |
Hyp |
Ref |
Expression |
1 |
|
offveq.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
offveq.2 |
⊢ ( 𝜑 → 𝐹 Fn 𝐴 ) |
3 |
|
offveq.3 |
⊢ ( 𝜑 → 𝐺 Fn 𝐴 ) |
4 |
|
offveq.4 |
⊢ ( 𝜑 → 𝐻 Fn 𝐴 ) |
5 |
|
offveq.5 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) = 𝐵 ) |
6 |
|
offveq.6 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑥 ) = 𝐶 ) |
7 |
|
offveq.7 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐵 𝑅 𝐶 ) = ( 𝐻 ‘ 𝑥 ) ) |
8 |
|
inidm |
⊢ ( 𝐴 ∩ 𝐴 ) = 𝐴 |
9 |
2 3 1 1 8
|
offn |
⊢ ( 𝜑 → ( 𝐹 ∘f 𝑅 𝐺 ) Fn 𝐴 ) |
10 |
2 3 1 1 8 5 6
|
ofval |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐹 ∘f 𝑅 𝐺 ) ‘ 𝑥 ) = ( 𝐵 𝑅 𝐶 ) ) |
11 |
10 7
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( 𝐹 ∘f 𝑅 𝐺 ) ‘ 𝑥 ) = ( 𝐻 ‘ 𝑥 ) ) |
12 |
9 4 11
|
eqfnfvd |
⊢ ( 𝜑 → ( 𝐹 ∘f 𝑅 𝐺 ) = 𝐻 ) |