Step |
Hyp |
Ref |
Expression |
1 |
|
f1f |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
2 |
1
|
frnd |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → ran 𝐹 ⊆ 𝐵 ) |
3 |
|
f1f |
⊢ ( 𝐺 : 𝐶 –1-1→ 𝐷 → 𝐺 : 𝐶 ⟶ 𝐷 ) |
4 |
3
|
frnd |
⊢ ( 𝐺 : 𝐶 –1-1→ 𝐷 → ran 𝐺 ⊆ 𝐷 ) |
5 |
|
unss12 |
⊢ ( ( ran 𝐹 ⊆ 𝐵 ∧ ran 𝐺 ⊆ 𝐷 ) → ( ran 𝐹 ∪ ran 𝐺 ) ⊆ ( 𝐵 ∪ 𝐷 ) ) |
6 |
2 4 5
|
syl2an |
⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐺 : 𝐶 –1-1→ 𝐷 ) → ( ran 𝐹 ∪ ran 𝐺 ) ⊆ ( 𝐵 ∪ 𝐷 ) ) |
7 |
|
f1f1orn |
⊢ ( 𝐹 : 𝐴 –1-1→ 𝐵 → 𝐹 : 𝐴 –1-1-onto→ ran 𝐹 ) |
8 |
|
f1f1orn |
⊢ ( 𝐺 : 𝐶 –1-1→ 𝐷 → 𝐺 : 𝐶 –1-1-onto→ ran 𝐺 ) |
9 |
7 8
|
anim12i |
⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐺 : 𝐶 –1-1→ 𝐷 ) → ( 𝐹 : 𝐴 –1-1-onto→ ran 𝐹 ∧ 𝐺 : 𝐶 –1-1-onto→ ran 𝐺 ) ) |
10 |
|
simprl |
⊢ ( ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐺 : 𝐶 –1-1→ 𝐷 ) ∧ ( ( 𝐴 ∩ 𝐶 ) = ∅ ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ) → ( 𝐴 ∩ 𝐶 ) = ∅ ) |
11 |
|
ss2in |
⊢ ( ( ran 𝐹 ⊆ 𝐵 ∧ ran 𝐺 ⊆ 𝐷 ) → ( ran 𝐹 ∩ ran 𝐺 ) ⊆ ( 𝐵 ∩ 𝐷 ) ) |
12 |
2 4 11
|
syl2an |
⊢ ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐺 : 𝐶 –1-1→ 𝐷 ) → ( ran 𝐹 ∩ ran 𝐺 ) ⊆ ( 𝐵 ∩ 𝐷 ) ) |
13 |
|
sseq0 |
⊢ ( ( ( ran 𝐹 ∩ ran 𝐺 ) ⊆ ( 𝐵 ∩ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) → ( ran 𝐹 ∩ ran 𝐺 ) = ∅ ) |
14 |
12 13
|
sylan |
⊢ ( ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐺 : 𝐶 –1-1→ 𝐷 ) ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) → ( ran 𝐹 ∩ ran 𝐺 ) = ∅ ) |
15 |
14
|
adantrl |
⊢ ( ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐺 : 𝐶 –1-1→ 𝐷 ) ∧ ( ( 𝐴 ∩ 𝐶 ) = ∅ ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ) → ( ran 𝐹 ∩ ran 𝐺 ) = ∅ ) |
16 |
10 15
|
jca |
⊢ ( ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐺 : 𝐶 –1-1→ 𝐷 ) ∧ ( ( 𝐴 ∩ 𝐶 ) = ∅ ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ) → ( ( 𝐴 ∩ 𝐶 ) = ∅ ∧ ( ran 𝐹 ∩ ran 𝐺 ) = ∅ ) ) |
17 |
|
f1oun |
⊢ ( ( ( 𝐹 : 𝐴 –1-1-onto→ ran 𝐹 ∧ 𝐺 : 𝐶 –1-1-onto→ ran 𝐺 ) ∧ ( ( 𝐴 ∩ 𝐶 ) = ∅ ∧ ( ran 𝐹 ∩ ran 𝐺 ) = ∅ ) ) → ( 𝐹 ∪ 𝐺 ) : ( 𝐴 ∪ 𝐶 ) –1-1-onto→ ( ran 𝐹 ∪ ran 𝐺 ) ) |
18 |
9 16 17
|
syl2an2r |
⊢ ( ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐺 : 𝐶 –1-1→ 𝐷 ) ∧ ( ( 𝐴 ∩ 𝐶 ) = ∅ ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ) → ( 𝐹 ∪ 𝐺 ) : ( 𝐴 ∪ 𝐶 ) –1-1-onto→ ( ran 𝐹 ∪ ran 𝐺 ) ) |
19 |
|
f1of1 |
⊢ ( ( 𝐹 ∪ 𝐺 ) : ( 𝐴 ∪ 𝐶 ) –1-1-onto→ ( ran 𝐹 ∪ ran 𝐺 ) → ( 𝐹 ∪ 𝐺 ) : ( 𝐴 ∪ 𝐶 ) –1-1→ ( ran 𝐹 ∪ ran 𝐺 ) ) |
20 |
18 19
|
syl |
⊢ ( ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐺 : 𝐶 –1-1→ 𝐷 ) ∧ ( ( 𝐴 ∩ 𝐶 ) = ∅ ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ) → ( 𝐹 ∪ 𝐺 ) : ( 𝐴 ∪ 𝐶 ) –1-1→ ( ran 𝐹 ∪ ran 𝐺 ) ) |
21 |
|
f1ss |
⊢ ( ( ( 𝐹 ∪ 𝐺 ) : ( 𝐴 ∪ 𝐶 ) –1-1→ ( ran 𝐹 ∪ ran 𝐺 ) ∧ ( ran 𝐹 ∪ ran 𝐺 ) ⊆ ( 𝐵 ∪ 𝐷 ) ) → ( 𝐹 ∪ 𝐺 ) : ( 𝐴 ∪ 𝐶 ) –1-1→ ( 𝐵 ∪ 𝐷 ) ) |
22 |
21
|
ancoms |
⊢ ( ( ( ran 𝐹 ∪ ran 𝐺 ) ⊆ ( 𝐵 ∪ 𝐷 ) ∧ ( 𝐹 ∪ 𝐺 ) : ( 𝐴 ∪ 𝐶 ) –1-1→ ( ran 𝐹 ∪ ran 𝐺 ) ) → ( 𝐹 ∪ 𝐺 ) : ( 𝐴 ∪ 𝐶 ) –1-1→ ( 𝐵 ∪ 𝐷 ) ) |
23 |
6 20 22
|
syl2an2r |
⊢ ( ( ( 𝐹 : 𝐴 –1-1→ 𝐵 ∧ 𝐺 : 𝐶 –1-1→ 𝐷 ) ∧ ( ( 𝐴 ∩ 𝐶 ) = ∅ ∧ ( 𝐵 ∩ 𝐷 ) = ∅ ) ) → ( 𝐹 ∪ 𝐺 ) : ( 𝐴 ∪ 𝐶 ) –1-1→ ( 𝐵 ∪ 𝐷 ) ) |