Step |
Hyp |
Ref |
Expression |
1 |
|
relco |
⊢ Rel ( 𝐴 ∘ 𝐵 ) |
2 |
|
ssrel3 |
⊢ ( Rel ( 𝐴 ∘ 𝐵 ) → ( ( 𝐴 ∘ 𝐵 ) ⊆ 𝐶 ↔ ∀ 𝑥 ∀ 𝑧 ( 𝑥 ( 𝐴 ∘ 𝐵 ) 𝑧 → 𝑥 𝐶 𝑧 ) ) ) |
3 |
1 2
|
ax-mp |
⊢ ( ( 𝐴 ∘ 𝐵 ) ⊆ 𝐶 ↔ ∀ 𝑥 ∀ 𝑧 ( 𝑥 ( 𝐴 ∘ 𝐵 ) 𝑧 → 𝑥 𝐶 𝑧 ) ) |
4 |
|
vex |
⊢ 𝑥 ∈ V |
5 |
|
vex |
⊢ 𝑧 ∈ V |
6 |
4 5
|
brco |
⊢ ( 𝑥 ( 𝐴 ∘ 𝐵 ) 𝑧 ↔ ∃ 𝑦 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) ) |
7 |
6
|
imbi1i |
⊢ ( ( 𝑥 ( 𝐴 ∘ 𝐵 ) 𝑧 → 𝑥 𝐶 𝑧 ) ↔ ( ∃ 𝑦 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) ) |
8 |
|
19.23v |
⊢ ( ∀ 𝑦 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) ↔ ( ∃ 𝑦 ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) ) |
9 |
7 8
|
bitr4i |
⊢ ( ( 𝑥 ( 𝐴 ∘ 𝐵 ) 𝑧 → 𝑥 𝐶 𝑧 ) ↔ ∀ 𝑦 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) ) |
10 |
9
|
albii |
⊢ ( ∀ 𝑧 ( 𝑥 ( 𝐴 ∘ 𝐵 ) 𝑧 → 𝑥 𝐶 𝑧 ) ↔ ∀ 𝑧 ∀ 𝑦 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) ) |
11 |
|
breq2 |
⊢ ( 𝑧 = 𝑤 → ( 𝑦 𝐴 𝑧 ↔ 𝑦 𝐴 𝑤 ) ) |
12 |
11
|
anbi2d |
⊢ ( 𝑧 = 𝑤 → ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) ↔ ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑤 ) ) ) |
13 |
|
breq2 |
⊢ ( 𝑧 = 𝑤 → ( 𝑥 𝐶 𝑧 ↔ 𝑥 𝐶 𝑤 ) ) |
14 |
12 13
|
imbi12d |
⊢ ( 𝑧 = 𝑤 → ( ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) ↔ ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑤 ) → 𝑥 𝐶 𝑤 ) ) ) |
15 |
|
breq2 |
⊢ ( 𝑦 = 𝑤 → ( 𝑥 𝐵 𝑦 ↔ 𝑥 𝐵 𝑤 ) ) |
16 |
|
breq1 |
⊢ ( 𝑦 = 𝑤 → ( 𝑦 𝐴 𝑧 ↔ 𝑤 𝐴 𝑧 ) ) |
17 |
15 16
|
anbi12d |
⊢ ( 𝑦 = 𝑤 → ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) ↔ ( 𝑥 𝐵 𝑤 ∧ 𝑤 𝐴 𝑧 ) ) ) |
18 |
17
|
imbi1d |
⊢ ( 𝑦 = 𝑤 → ( ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) ↔ ( ( 𝑥 𝐵 𝑤 ∧ 𝑤 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) ) ) |
19 |
14 18
|
alcomw |
⊢ ( ∀ 𝑧 ∀ 𝑦 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) ↔ ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) ) |
20 |
10 19
|
bitri |
⊢ ( ∀ 𝑧 ( 𝑥 ( 𝐴 ∘ 𝐵 ) 𝑧 → 𝑥 𝐶 𝑧 ) ↔ ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) ) |
21 |
20
|
albii |
⊢ ( ∀ 𝑥 ∀ 𝑧 ( 𝑥 ( 𝐴 ∘ 𝐵 ) 𝑧 → 𝑥 𝐶 𝑧 ) ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) ) |
22 |
3 21
|
bitri |
⊢ ( ( 𝐴 ∘ 𝐵 ) ⊆ 𝐶 ↔ ∀ 𝑥 ∀ 𝑦 ∀ 𝑧 ( ( 𝑥 𝐵 𝑦 ∧ 𝑦 𝐴 𝑧 ) → 𝑥 𝐶 𝑧 ) ) |