Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ( Base ‘ 𝑅 ) = ( Base ‘ 𝑅 ) |
2 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
3 |
1 2
|
rhmf |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) ) |
4 |
|
frel |
⊢ ( 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) → Rel 𝐹 ) |
5 |
|
dfrel2 |
⊢ ( Rel 𝐹 ↔ ◡ ◡ 𝐹 = 𝐹 ) |
6 |
4 5
|
sylib |
⊢ ( 𝐹 : ( Base ‘ 𝑅 ) ⟶ ( Base ‘ 𝑆 ) → ◡ ◡ 𝐹 = 𝐹 ) |
7 |
3 6
|
syl |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ◡ ◡ 𝐹 = 𝐹 ) |
8 |
|
id |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) |
9 |
7 8
|
eqeltrd |
⊢ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) → ◡ ◡ 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) |
10 |
9
|
anim1ci |
⊢ ( ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ◡ 𝐹 ∈ ( 𝑆 RingHom 𝑅 ) ) → ( ◡ 𝐹 ∈ ( 𝑆 RingHom 𝑅 ) ∧ ◡ ◡ 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) ) |
11 |
|
isrim0 |
⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) ↔ ( 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ∧ ◡ 𝐹 ∈ ( 𝑆 RingHom 𝑅 ) ) ) |
12 |
|
isrim0 |
⊢ ( ◡ 𝐹 ∈ ( 𝑆 RingIso 𝑅 ) ↔ ( ◡ 𝐹 ∈ ( 𝑆 RingHom 𝑅 ) ∧ ◡ ◡ 𝐹 ∈ ( 𝑅 RingHom 𝑆 ) ) ) |
13 |
10 11 12
|
3imtr4i |
⊢ ( 𝐹 ∈ ( 𝑅 RingIso 𝑆 ) → ◡ 𝐹 ∈ ( 𝑆 RingIso 𝑅 ) ) |