Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → 𝐹 Fn 𝐴 ) |
2 |
|
simpl |
⊢ ( ( 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → 𝐵 ⊆ 𝐴 ) |
3 |
|
simpr |
⊢ ( ( 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → 𝐶 ⊆ 𝐴 ) |
4 |
2 3
|
unssd |
⊢ ( ( 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝐵 ∪ 𝐶 ) ⊆ 𝐴 ) |
5 |
4
|
3adant1 |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝐵 ∪ 𝐶 ) ⊆ 𝐴 ) |
6 |
1 5
|
fvelimabd |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ ( 𝐵 ∪ 𝐶 ) ) ↔ ∃ 𝑥 ∈ ( 𝐵 ∪ 𝐶 ) ( 𝐹 ‘ 𝑥 ) = 𝑦 ) ) |
7 |
|
rexun |
⊢ ( ∃ 𝑥 ∈ ( 𝐵 ∪ 𝐶 ) ( 𝐹 ‘ 𝑥 ) = 𝑦 ↔ ( ∃ 𝑥 ∈ 𝐵 ( 𝐹 ‘ 𝑥 ) = 𝑦 ∨ ∃ 𝑥 ∈ 𝐶 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) ) |
8 |
6 7
|
bitrdi |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ ( 𝐵 ∪ 𝐶 ) ) ↔ ( ∃ 𝑥 ∈ 𝐵 ( 𝐹 ‘ 𝑥 ) = 𝑦 ∨ ∃ 𝑥 ∈ 𝐶 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) ) ) |
9 |
|
fvelimab |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐵 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) ) |
10 |
9
|
3adant3 |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐵 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) ) |
11 |
|
fvelimab |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐶 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) ) |
12 |
11
|
3adant2 |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ 𝐶 ) ↔ ∃ 𝑥 ∈ 𝐶 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) ) |
13 |
10 12
|
orbi12d |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( ( 𝑦 ∈ ( 𝐹 “ 𝐵 ) ∨ 𝑦 ∈ ( 𝐹 “ 𝐶 ) ) ↔ ( ∃ 𝑥 ∈ 𝐵 ( 𝐹 ‘ 𝑥 ) = 𝑦 ∨ ∃ 𝑥 ∈ 𝐶 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) ) ) |
14 |
8 13
|
bitr4d |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ ( 𝐵 ∪ 𝐶 ) ) ↔ ( 𝑦 ∈ ( 𝐹 “ 𝐵 ) ∨ 𝑦 ∈ ( 𝐹 “ 𝐶 ) ) ) ) |
15 |
|
elun |
⊢ ( 𝑦 ∈ ( ( 𝐹 “ 𝐵 ) ∪ ( 𝐹 “ 𝐶 ) ) ↔ ( 𝑦 ∈ ( 𝐹 “ 𝐵 ) ∨ 𝑦 ∈ ( 𝐹 “ 𝐶 ) ) ) |
16 |
14 15
|
bitr4di |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝑦 ∈ ( 𝐹 “ ( 𝐵 ∪ 𝐶 ) ) ↔ 𝑦 ∈ ( ( 𝐹 “ 𝐵 ) ∪ ( 𝐹 “ 𝐶 ) ) ) ) |
17 |
16
|
eqrdv |
⊢ ( ( 𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ⊆ 𝐴 ) → ( 𝐹 “ ( 𝐵 ∪ 𝐶 ) ) = ( ( 𝐹 “ 𝐵 ) ∪ ( 𝐹 “ 𝐶 ) ) ) |