Step |
Hyp |
Ref |
Expression |
1 |
|
df-com2 |
⊢ Com2 = { 〈 𝑥 , 𝑦 〉 ∣ ∀ 𝑎 ∈ ran 𝑥 ∀ 𝑏 ∈ ran 𝑥 ( 𝑎 𝑦 𝑏 ) = ( 𝑏 𝑦 𝑎 ) } |
2 |
1
|
a1i |
⊢ ( ( 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵 ) → Com2 = { 〈 𝑥 , 𝑦 〉 ∣ ∀ 𝑎 ∈ ran 𝑥 ∀ 𝑏 ∈ ran 𝑥 ( 𝑎 𝑦 𝑏 ) = ( 𝑏 𝑦 𝑎 ) } ) |
3 |
2
|
eleq2d |
⊢ ( ( 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵 ) → ( 〈 𝐺 , 𝐻 〉 ∈ Com2 ↔ 〈 𝐺 , 𝐻 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ∀ 𝑎 ∈ ran 𝑥 ∀ 𝑏 ∈ ran 𝑥 ( 𝑎 𝑦 𝑏 ) = ( 𝑏 𝑦 𝑎 ) } ) ) |
4 |
|
rneq |
⊢ ( 𝑥 = 𝐺 → ran 𝑥 = ran 𝐺 ) |
5 |
4
|
raleqdv |
⊢ ( 𝑥 = 𝐺 → ( ∀ 𝑏 ∈ ran 𝑥 ( 𝑎 𝑦 𝑏 ) = ( 𝑏 𝑦 𝑎 ) ↔ ∀ 𝑏 ∈ ran 𝐺 ( 𝑎 𝑦 𝑏 ) = ( 𝑏 𝑦 𝑎 ) ) ) |
6 |
4 5
|
raleqbidv |
⊢ ( 𝑥 = 𝐺 → ( ∀ 𝑎 ∈ ran 𝑥 ∀ 𝑏 ∈ ran 𝑥 ( 𝑎 𝑦 𝑏 ) = ( 𝑏 𝑦 𝑎 ) ↔ ∀ 𝑎 ∈ ran 𝐺 ∀ 𝑏 ∈ ran 𝐺 ( 𝑎 𝑦 𝑏 ) = ( 𝑏 𝑦 𝑎 ) ) ) |
7 |
|
oveq |
⊢ ( 𝑦 = 𝐻 → ( 𝑎 𝑦 𝑏 ) = ( 𝑎 𝐻 𝑏 ) ) |
8 |
|
oveq |
⊢ ( 𝑦 = 𝐻 → ( 𝑏 𝑦 𝑎 ) = ( 𝑏 𝐻 𝑎 ) ) |
9 |
7 8
|
eqeq12d |
⊢ ( 𝑦 = 𝐻 → ( ( 𝑎 𝑦 𝑏 ) = ( 𝑏 𝑦 𝑎 ) ↔ ( 𝑎 𝐻 𝑏 ) = ( 𝑏 𝐻 𝑎 ) ) ) |
10 |
9
|
2ralbidv |
⊢ ( 𝑦 = 𝐻 → ( ∀ 𝑎 ∈ ran 𝐺 ∀ 𝑏 ∈ ran 𝐺 ( 𝑎 𝑦 𝑏 ) = ( 𝑏 𝑦 𝑎 ) ↔ ∀ 𝑎 ∈ ran 𝐺 ∀ 𝑏 ∈ ran 𝐺 ( 𝑎 𝐻 𝑏 ) = ( 𝑏 𝐻 𝑎 ) ) ) |
11 |
6 10
|
opelopabg |
⊢ ( ( 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵 ) → ( 〈 𝐺 , 𝐻 〉 ∈ { 〈 𝑥 , 𝑦 〉 ∣ ∀ 𝑎 ∈ ran 𝑥 ∀ 𝑏 ∈ ran 𝑥 ( 𝑎 𝑦 𝑏 ) = ( 𝑏 𝑦 𝑎 ) } ↔ ∀ 𝑎 ∈ ran 𝐺 ∀ 𝑏 ∈ ran 𝐺 ( 𝑎 𝐻 𝑏 ) = ( 𝑏 𝐻 𝑎 ) ) ) |
12 |
3 11
|
bitrd |
⊢ ( ( 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐵 ) → ( 〈 𝐺 , 𝐻 〉 ∈ Com2 ↔ ∀ 𝑎 ∈ ran 𝐺 ∀ 𝑏 ∈ ran 𝐺 ( 𝑎 𝐻 𝑏 ) = ( 𝑏 𝐻 𝑎 ) ) ) |