Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ( Base ‘ 𝑆 ) = ( Base ‘ 𝑆 ) |
2 |
|
eqid |
⊢ ( Base ‘ 𝑇 ) = ( Base ‘ 𝑇 ) |
3 |
1 2
|
lmhmf |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) ) |
4 |
|
frel |
⊢ ( 𝐹 : ( Base ‘ 𝑆 ) ⟶ ( Base ‘ 𝑇 ) → Rel 𝐹 ) |
5 |
3 4
|
syl |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → Rel 𝐹 ) |
6 |
|
dfrel2 |
⊢ ( Rel 𝐹 ↔ ◡ ◡ 𝐹 = 𝐹 ) |
7 |
5 6
|
sylib |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → ◡ ◡ 𝐹 = 𝐹 ) |
8 |
|
id |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) |
9 |
7 8
|
eqeltrd |
⊢ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) → ◡ ◡ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) |
10 |
9
|
anim1ci |
⊢ ( ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ ◡ 𝐹 ∈ ( 𝑇 LMHom 𝑆 ) ) → ( ◡ 𝐹 ∈ ( 𝑇 LMHom 𝑆 ) ∧ ◡ ◡ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ) |
11 |
|
islmim2 |
⊢ ( 𝐹 ∈ ( 𝑆 LMIso 𝑇 ) ↔ ( 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ∧ ◡ 𝐹 ∈ ( 𝑇 LMHom 𝑆 ) ) ) |
12 |
|
islmim2 |
⊢ ( ◡ 𝐹 ∈ ( 𝑇 LMIso 𝑆 ) ↔ ( ◡ 𝐹 ∈ ( 𝑇 LMHom 𝑆 ) ∧ ◡ ◡ 𝐹 ∈ ( 𝑆 LMHom 𝑇 ) ) ) |
13 |
10 11 12
|
3imtr4i |
⊢ ( 𝐹 ∈ ( 𝑆 LMIso 𝑇 ) → ◡ 𝐹 ∈ ( 𝑇 LMIso 𝑆 ) ) |