Step |
Hyp |
Ref |
Expression |
1 |
|
uncom |
⊢ ( 𝐹 ∪ 𝐺 ) = ( 𝐺 ∪ 𝐹 ) |
2 |
1
|
reseq1i |
⊢ ( ( 𝐹 ∪ 𝐺 ) ↾ 𝐵 ) = ( ( 𝐺 ∪ 𝐹 ) ↾ 𝐵 ) |
3 |
|
simp2 |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → 𝐺 Fn 𝐵 ) |
4 |
|
simp1 |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → 𝐹 Fn 𝐴 ) |
5 |
|
incom |
⊢ ( 𝐴 ∩ 𝐵 ) = ( 𝐵 ∩ 𝐴 ) |
6 |
|
simp3 |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( 𝐴 ∩ 𝐵 ) = ∅ ) |
7 |
5 6
|
eqtr3id |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( 𝐵 ∩ 𝐴 ) = ∅ ) |
8 |
|
fnunres1 |
⊢ ( ( 𝐺 Fn 𝐵 ∧ 𝐹 Fn 𝐴 ∧ ( 𝐵 ∩ 𝐴 ) = ∅ ) → ( ( 𝐺 ∪ 𝐹 ) ↾ 𝐵 ) = 𝐺 ) |
9 |
3 4 7 8
|
syl3anc |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( ( 𝐺 ∪ 𝐹 ) ↾ 𝐵 ) = 𝐺 ) |
10 |
2 9
|
syl5eq |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐵 ∧ ( 𝐴 ∩ 𝐵 ) = ∅ ) → ( ( 𝐹 ∪ 𝐺 ) ↾ 𝐵 ) = 𝐺 ) |