Step |
Hyp |
Ref |
Expression |
1 |
|
ofc12.1 |
⊢ ( 𝜑 → 𝐴 ∈ 𝑉 ) |
2 |
|
ofc12.2 |
⊢ ( 𝜑 → 𝐵 ∈ 𝑊 ) |
3 |
|
ofc12.3 |
⊢ ( 𝜑 → 𝐶 ∈ 𝑋 ) |
4 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝑊 ) |
5 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐶 ∈ 𝑋 ) |
6 |
|
fconstmpt |
⊢ ( 𝐴 × { 𝐵 } ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) |
7 |
6
|
a1i |
⊢ ( 𝜑 → ( 𝐴 × { 𝐵 } ) = ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ) |
8 |
|
fconstmpt |
⊢ ( 𝐴 × { 𝐶 } ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) |
9 |
8
|
a1i |
⊢ ( 𝜑 → ( 𝐴 × { 𝐶 } ) = ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ) |
10 |
1 4 5 7 9
|
offval2 |
⊢ ( 𝜑 → ( ( 𝐴 × { 𝐵 } ) ∘f 𝑅 ( 𝐴 × { 𝐶 } ) ) = ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 𝑅 𝐶 ) ) ) |
11 |
|
fconstmpt |
⊢ ( 𝐴 × { ( 𝐵 𝑅 𝐶 ) } ) = ( 𝑥 ∈ 𝐴 ↦ ( 𝐵 𝑅 𝐶 ) ) |
12 |
10 11
|
eqtr4di |
⊢ ( 𝜑 → ( ( 𝐴 × { 𝐵 } ) ∘f 𝑅 ( 𝐴 × { 𝐶 } ) ) = ( 𝐴 × { ( 𝐵 𝑅 𝐶 ) } ) ) |