Step |
Hyp |
Ref |
Expression |
1 |
|
natrcl.1 |
⊢ 𝑁 = ( 𝐶 Nat 𝐷 ) |
2 |
|
nat1st2nd.2 |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝐹 𝑁 𝐺 ) ) |
3 |
|
relfunc |
⊢ Rel ( 𝐶 Func 𝐷 ) |
4 |
1
|
natrcl |
⊢ ( 𝐴 ∈ ( 𝐹 𝑁 𝐺 ) → ( 𝐹 ∈ ( 𝐶 Func 𝐷 ) ∧ 𝐺 ∈ ( 𝐶 Func 𝐷 ) ) ) |
5 |
2 4
|
syl |
⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝐶 Func 𝐷 ) ∧ 𝐺 ∈ ( 𝐶 Func 𝐷 ) ) ) |
6 |
5
|
simpld |
⊢ ( 𝜑 → 𝐹 ∈ ( 𝐶 Func 𝐷 ) ) |
7 |
|
1st2nd |
⊢ ( ( Rel ( 𝐶 Func 𝐷 ) ∧ 𝐹 ∈ ( 𝐶 Func 𝐷 ) ) → 𝐹 = 〈 ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) 〉 ) |
8 |
3 6 7
|
sylancr |
⊢ ( 𝜑 → 𝐹 = 〈 ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) 〉 ) |
9 |
5
|
simprd |
⊢ ( 𝜑 → 𝐺 ∈ ( 𝐶 Func 𝐷 ) ) |
10 |
|
1st2nd |
⊢ ( ( Rel ( 𝐶 Func 𝐷 ) ∧ 𝐺 ∈ ( 𝐶 Func 𝐷 ) ) → 𝐺 = 〈 ( 1st ‘ 𝐺 ) , ( 2nd ‘ 𝐺 ) 〉 ) |
11 |
3 9 10
|
sylancr |
⊢ ( 𝜑 → 𝐺 = 〈 ( 1st ‘ 𝐺 ) , ( 2nd ‘ 𝐺 ) 〉 ) |
12 |
8 11
|
oveq12d |
⊢ ( 𝜑 → ( 𝐹 𝑁 𝐺 ) = ( 〈 ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) 〉 𝑁 〈 ( 1st ‘ 𝐺 ) , ( 2nd ‘ 𝐺 ) 〉 ) ) |
13 |
2 12
|
eleqtrd |
⊢ ( 𝜑 → 𝐴 ∈ ( 〈 ( 1st ‘ 𝐹 ) , ( 2nd ‘ 𝐹 ) 〉 𝑁 〈 ( 1st ‘ 𝐺 ) , ( 2nd ‘ 𝐺 ) 〉 ) ) |