Step |
Hyp |
Ref |
Expression |
1 |
|
coires1 |
⊢ ( 𝐹 ∘ ( I ↾ dom ( 𝐹 ↾ 𝑋 ) ) ) = ( 𝐹 ↾ dom ( 𝐹 ↾ 𝑋 ) ) |
2 |
|
resdmres |
⊢ ( 𝐹 ↾ dom ( 𝐹 ↾ 𝑋 ) ) = ( 𝐹 ↾ 𝑋 ) |
3 |
1 2
|
eqtri |
⊢ ( 𝐹 ∘ ( I ↾ dom ( 𝐹 ↾ 𝑋 ) ) ) = ( 𝐹 ↾ 𝑋 ) |
4 |
|
f1oi |
⊢ ( I ↾ dom ( 𝐹 ↾ 𝑋 ) ) : dom ( 𝐹 ↾ 𝑋 ) –1-1-onto→ dom ( 𝐹 ↾ 𝑋 ) |
5 |
|
f1of1 |
⊢ ( ( I ↾ dom ( 𝐹 ↾ 𝑋 ) ) : dom ( 𝐹 ↾ 𝑋 ) –1-1-onto→ dom ( 𝐹 ↾ 𝑋 ) → ( I ↾ dom ( 𝐹 ↾ 𝑋 ) ) : dom ( 𝐹 ↾ 𝑋 ) –1-1→ dom ( 𝐹 ↾ 𝑋 ) ) |
6 |
4 5
|
ax-mp |
⊢ ( I ↾ dom ( 𝐹 ↾ 𝑋 ) ) : dom ( 𝐹 ↾ 𝑋 ) –1-1→ dom ( 𝐹 ↾ 𝑋 ) |
7 |
|
resss |
⊢ ( 𝐹 ↾ 𝑋 ) ⊆ 𝐹 |
8 |
|
dmss |
⊢ ( ( 𝐹 ↾ 𝑋 ) ⊆ 𝐹 → dom ( 𝐹 ↾ 𝑋 ) ⊆ dom 𝐹 ) |
9 |
7 8
|
ax-mp |
⊢ dom ( 𝐹 ↾ 𝑋 ) ⊆ dom 𝐹 |
10 |
|
f1ss |
⊢ ( ( ( I ↾ dom ( 𝐹 ↾ 𝑋 ) ) : dom ( 𝐹 ↾ 𝑋 ) –1-1→ dom ( 𝐹 ↾ 𝑋 ) ∧ dom ( 𝐹 ↾ 𝑋 ) ⊆ dom 𝐹 ) → ( I ↾ dom ( 𝐹 ↾ 𝑋 ) ) : dom ( 𝐹 ↾ 𝑋 ) –1-1→ dom 𝐹 ) |
11 |
6 9 10
|
mp2an |
⊢ ( I ↾ dom ( 𝐹 ↾ 𝑋 ) ) : dom ( 𝐹 ↾ 𝑋 ) –1-1→ dom 𝐹 |
12 |
|
f1lindf |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ∧ ( I ↾ dom ( 𝐹 ↾ 𝑋 ) ) : dom ( 𝐹 ↾ 𝑋 ) –1-1→ dom 𝐹 ) → ( 𝐹 ∘ ( I ↾ dom ( 𝐹 ↾ 𝑋 ) ) ) LIndF 𝑊 ) |
13 |
11 12
|
mp3an3 |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ) → ( 𝐹 ∘ ( I ↾ dom ( 𝐹 ↾ 𝑋 ) ) ) LIndF 𝑊 ) |
14 |
3 13
|
eqbrtrrid |
⊢ ( ( 𝑊 ∈ LMod ∧ 𝐹 LIndF 𝑊 ) → ( 𝐹 ↾ 𝑋 ) LIndF 𝑊 ) |