Step |
Hyp |
Ref |
Expression |
1 |
|
eluni |
⊢ ( 𝑦 ∈ ∪ 𝐴 ↔ ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) ) |
2 |
1
|
imbi1i |
⊢ ( ( 𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵 ) ↔ ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) ) |
3 |
|
19.23v |
⊢ ( ∀ 𝑥 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) ↔ ( ∃ 𝑥 ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) ) |
4 |
2 3
|
bitr4i |
⊢ ( ( 𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵 ) ↔ ∀ 𝑥 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) ) |
5 |
4
|
albii |
⊢ ( ∀ 𝑦 ( 𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵 ) ↔ ∀ 𝑦 ∀ 𝑥 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) ) |
6 |
|
elequ1 |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∈ 𝑥 ↔ 𝑧 ∈ 𝑥 ) ) |
7 |
6
|
anbi1d |
⊢ ( 𝑦 = 𝑧 → ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) ↔ ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) ) ) |
8 |
|
eleq1w |
⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∈ 𝐵 ↔ 𝑧 ∈ 𝐵 ) ) |
9 |
7 8
|
imbi12d |
⊢ ( 𝑦 = 𝑧 → ( ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) ↔ ( ( 𝑧 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑧 ∈ 𝐵 ) ) ) |
10 |
|
elequ2 |
⊢ ( 𝑥 = 𝑧 → ( 𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑧 ) ) |
11 |
|
eleq1w |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴 ) ) |
12 |
10 11
|
anbi12d |
⊢ ( 𝑥 = 𝑧 → ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) ↔ ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴 ) ) ) |
13 |
12
|
imbi1d |
⊢ ( 𝑥 = 𝑧 → ( ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) ↔ ( ( 𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) ) ) |
14 |
9 13
|
alcomw |
⊢ ( ∀ 𝑦 ∀ 𝑥 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) ) |
15 |
|
19.21v |
⊢ ( ∀ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵 ) ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵 ) ) ) |
16 |
|
impexp |
⊢ ( ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) ↔ ( 𝑦 ∈ 𝑥 → ( 𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐵 ) ) ) |
17 |
|
bi2.04 |
⊢ ( ( 𝑦 ∈ 𝑥 → ( 𝑥 ∈ 𝐴 → 𝑦 ∈ 𝐵 ) ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵 ) ) ) |
18 |
16 17
|
bitri |
⊢ ( ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵 ) ) ) |
19 |
18
|
albii |
⊢ ( ∀ 𝑦 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) ↔ ∀ 𝑦 ( 𝑥 ∈ 𝐴 → ( 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵 ) ) ) |
20 |
|
dfss2 |
⊢ ( 𝑥 ⊆ 𝐵 ↔ ∀ 𝑦 ( 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵 ) ) |
21 |
20
|
imbi2i |
⊢ ( ( 𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 → ∀ 𝑦 ( 𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐵 ) ) ) |
22 |
15 19 21
|
3bitr4i |
⊢ ( ∀ 𝑦 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵 ) ) |
23 |
22
|
albii |
⊢ ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵 ) ) |
24 |
14 23
|
bitri |
⊢ ( ∀ 𝑦 ∀ 𝑥 ( ( 𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴 ) → 𝑦 ∈ 𝐵 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵 ) ) |
25 |
5 24
|
bitri |
⊢ ( ∀ 𝑦 ( 𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵 ) ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵 ) ) |
26 |
|
dfss2 |
⊢ ( ∪ 𝐴 ⊆ 𝐵 ↔ ∀ 𝑦 ( 𝑦 ∈ ∪ 𝐴 → 𝑦 ∈ 𝐵 ) ) |
27 |
|
df-ral |
⊢ ( ∀ 𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝑥 ⊆ 𝐵 ) ) |
28 |
25 26 27
|
3bitr4i |
⊢ ( ∪ 𝐴 ⊆ 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵 ) |