Step |
Hyp |
Ref |
Expression |
1 |
|
uniiun |
⊢ ∪ 𝐵 = ∪ 𝑦 ∈ 𝐵 𝑦 |
2 |
|
foelrni |
⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ∃ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑦 ) |
3 |
|
eqimss2 |
⊢ ( ( 𝐹 ‘ 𝑥 ) = 𝑦 → 𝑦 ⊆ ( 𝐹 ‘ 𝑥 ) ) |
4 |
3
|
reximi |
⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) = 𝑦 → ∃ 𝑥 ∈ 𝐴 𝑦 ⊆ ( 𝐹 ‘ 𝑥 ) ) |
5 |
2 4
|
syl |
⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝑦 ∈ 𝐵 ) → ∃ 𝑥 ∈ 𝐴 𝑦 ⊆ ( 𝐹 ‘ 𝑥 ) ) |
6 |
5
|
ralrimiva |
⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ∀ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐴 𝑦 ⊆ ( 𝐹 ‘ 𝑥 ) ) |
7 |
|
iunss2 |
⊢ ( ∀ 𝑦 ∈ 𝐵 ∃ 𝑥 ∈ 𝐴 𝑦 ⊆ ( 𝐹 ‘ 𝑥 ) → ∪ 𝑦 ∈ 𝐵 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ) |
8 |
6 7
|
syl |
⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ∪ 𝑦 ∈ 𝐵 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ) |
9 |
|
fof |
⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐵 ) |
10 |
9
|
ffvelrnda |
⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) |
11 |
|
ssidd |
⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) |
12 |
|
sseq2 |
⊢ ( 𝑦 = ( 𝐹 ‘ 𝑥 ) → ( ( 𝐹 ‘ 𝑥 ) ⊆ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) ) |
13 |
12
|
rspcev |
⊢ ( ( ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ∧ ( 𝐹 ‘ 𝑥 ) ⊆ ( 𝐹 ‘ 𝑥 ) ) → ∃ 𝑦 ∈ 𝐵 ( 𝐹 ‘ 𝑥 ) ⊆ 𝑦 ) |
14 |
10 11 13
|
syl2anc |
⊢ ( ( 𝐹 : 𝐴 –onto→ 𝐵 ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑦 ∈ 𝐵 ( 𝐹 ‘ 𝑥 ) ⊆ 𝑦 ) |
15 |
14
|
ralrimiva |
⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝐹 ‘ 𝑥 ) ⊆ 𝑦 ) |
16 |
|
iunss2 |
⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 ( 𝐹 ‘ 𝑥 ) ⊆ 𝑦 → ∪ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦 ) |
17 |
15 16
|
syl |
⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ∪ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ⊆ ∪ 𝑦 ∈ 𝐵 𝑦 ) |
18 |
8 17
|
eqssd |
⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ∪ 𝑦 ∈ 𝐵 𝑦 = ∪ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ) |
19 |
1 18
|
syl5eq |
⊢ ( 𝐹 : 𝐴 –onto→ 𝐵 → ∪ 𝐵 = ∪ 𝑥 ∈ 𝐴 ( 𝐹 ‘ 𝑥 ) ) |