Step |
Hyp |
Ref |
Expression |
1 |
|
df-fn |
⊢ ( 𝐹 Fn 𝐴 ↔ ( Fun 𝐹 ∧ dom 𝐹 = 𝐴 ) ) |
2 |
|
df-fn |
⊢ ( 𝐺 Fn 𝐵 ↔ ( Fun 𝐺 ∧ dom 𝐺 = 𝐵 ) ) |
3 |
|
ineq12 |
⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → ( dom 𝐹 ∩ dom 𝐺 ) = ( 𝐴 ∩ 𝐵 ) ) |
4 |
3
|
eqeq1d |
⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → ( ( dom 𝐹 ∩ dom 𝐺 ) = ∅ ↔ ( 𝐴 ∩ 𝐵 ) = ∅ ) ) |
5 |
4
|
anbi2d |
⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → ( ( ( Fun 𝐹 ∧ Fun 𝐺 ) ∧ ( dom 𝐹 ∩ dom 𝐺 ) = ∅ ) ↔ ( ( Fun 𝐹 ∧ Fun 𝐺 ) ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) ) ) |
6 |
|
funun |
⊢ ( ( ( Fun 𝐹 ∧ Fun 𝐺 ) ∧ ( dom 𝐹 ∩ dom 𝐺 ) = ∅ ) → Fun ( 𝐹 ∪ 𝐺 ) ) |
7 |
5 6
|
syl6bir |
⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → ( ( ( Fun 𝐹 ∧ Fun 𝐺 ) ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → Fun ( 𝐹 ∪ 𝐺 ) ) ) |
8 |
|
dmun |
⊢ dom ( 𝐹 ∪ 𝐺 ) = ( dom 𝐹 ∪ dom 𝐺 ) |
9 |
|
uneq12 |
⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → ( dom 𝐹 ∪ dom 𝐺 ) = ( 𝐴 ∪ 𝐵 ) ) |
10 |
8 9
|
eqtrid |
⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → dom ( 𝐹 ∪ 𝐺 ) = ( 𝐴 ∪ 𝐵 ) ) |
11 |
7 10
|
jctird |
⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → ( ( ( Fun 𝐹 ∧ Fun 𝐺 ) ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( Fun ( 𝐹 ∪ 𝐺 ) ∧ dom ( 𝐹 ∪ 𝐺 ) = ( 𝐴 ∪ 𝐵 ) ) ) ) |
12 |
|
df-fn |
⊢ ( ( 𝐹 ∪ 𝐺 ) Fn ( 𝐴 ∪ 𝐵 ) ↔ ( Fun ( 𝐹 ∪ 𝐺 ) ∧ dom ( 𝐹 ∪ 𝐺 ) = ( 𝐴 ∪ 𝐵 ) ) ) |
13 |
11 12
|
syl6ibr |
⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → ( ( ( Fun 𝐹 ∧ Fun 𝐺 ) ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( 𝐹 ∪ 𝐺 ) Fn ( 𝐴 ∪ 𝐵 ) ) ) |
14 |
13
|
expd |
⊢ ( ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) → ( ( Fun 𝐹 ∧ Fun 𝐺 ) → ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( 𝐹 ∪ 𝐺 ) Fn ( 𝐴 ∪ 𝐵 ) ) ) ) |
15 |
14
|
impcom |
⊢ ( ( ( Fun 𝐹 ∧ Fun 𝐺 ) ∧ ( dom 𝐹 = 𝐴 ∧ dom 𝐺 = 𝐵 ) ) → ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( 𝐹 ∪ 𝐺 ) Fn ( 𝐴 ∪ 𝐵 ) ) ) |
16 |
15
|
an4s |
⊢ ( ( ( Fun 𝐹 ∧ dom 𝐹 = 𝐴 ) ∧ ( Fun 𝐺 ∧ dom 𝐺 = 𝐵 ) ) → ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( 𝐹 ∪ 𝐺 ) Fn ( 𝐴 ∪ 𝐵 ) ) ) |
17 |
1 2 16
|
syl2anb |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ) → ( ( 𝐴 ∩ 𝐵 ) = ∅ → ( 𝐹 ∪ 𝐺 ) Fn ( 𝐴 ∪ 𝐵 ) ) ) |
18 |
17
|
imp |
⊢ ( ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ) ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( 𝐹 ∪ 𝐺 ) Fn ( 𝐴 ∪ 𝐵 ) ) |