Step |
Hyp |
Ref |
Expression |
1 |
|
f1of1 |
⊢ ( 𝐺 : 𝐶 –1-1-onto→ 𝐷 → 𝐺 : 𝐶 –1-1→ 𝐷 ) |
2 |
1
|
anim1i |
⊢ ( ( 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ( 𝐺 : 𝐶 –1-1→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) ) |
3 |
2
|
3adant1 |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ( 𝐺 : 𝐶 –1-1→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) ) |
4 |
|
f1ores |
⊢ ( ( 𝐺 : 𝐶 –1-1→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ( 𝐺 ↾ 𝐵 ) : 𝐵 –1-1-onto→ ( 𝐺 “ 𝐵 ) ) |
5 |
3 4
|
syl |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ( 𝐺 ↾ 𝐵 ) : 𝐵 –1-1-onto→ ( 𝐺 “ 𝐵 ) ) |
6 |
|
simp1 |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) |
7 |
|
f1oco |
⊢ ( ( ( 𝐺 ↾ 𝐵 ) : 𝐵 –1-1-onto→ ( 𝐺 “ 𝐵 ) ∧ 𝐹 : 𝐴 –1-1-onto→ 𝐵 ) → ( ( 𝐺 ↾ 𝐵 ) ∘ 𝐹 ) : 𝐴 –1-1-onto→ ( 𝐺 “ 𝐵 ) ) |
8 |
5 6 7
|
syl2anc |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ( ( 𝐺 ↾ 𝐵 ) ∘ 𝐹 ) : 𝐴 –1-1-onto→ ( 𝐺 “ 𝐵 ) ) |
9 |
|
f1ofo |
⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → 𝐹 : 𝐴 –onto→ 𝐵 ) |
10 |
|
forn |
⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ran 𝐹 = 𝐵 ) |
11 |
9 10
|
syl |
⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ran 𝐹 = 𝐵 ) |
12 |
11
|
eqimssd |
⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ran 𝐹 ⊆ 𝐵 ) |
13 |
12
|
3ad2ant1 |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ran 𝐹 ⊆ 𝐵 ) |
14 |
|
cores |
⊢ ( ran 𝐹 ⊆ 𝐵 → ( ( 𝐺 ↾ 𝐵 ) ∘ 𝐹 ) = ( 𝐺 ∘ 𝐹 ) ) |
15 |
14
|
eqcomd |
⊢ ( ran 𝐹 ⊆ 𝐵 → ( 𝐺 ∘ 𝐹 ) = ( ( 𝐺 ↾ 𝐵 ) ∘ 𝐹 ) ) |
16 |
13 15
|
syl |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ( 𝐺 ∘ 𝐹 ) = ( ( 𝐺 ↾ 𝐵 ) ∘ 𝐹 ) ) |
17 |
16
|
f1oeq1d |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ( ( 𝐺 ∘ 𝐹 ) : 𝐴 –1-1-onto→ ( 𝐺 “ 𝐵 ) ↔ ( ( 𝐺 ↾ 𝐵 ) ∘ 𝐹 ) : 𝐴 –1-1-onto→ ( 𝐺 “ 𝐵 ) ) ) |
18 |
8 17
|
mpbird |
⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ 𝐺 : 𝐶 –1-1-onto→ 𝐷 ∧ 𝐵 ⊆ 𝐶 ) → ( 𝐺 ∘ 𝐹 ) : 𝐴 –1-1-onto→ ( 𝐺 “ 𝐵 ) ) |