Step |
Hyp |
Ref |
Expression |
1 |
|
id |
⊢ ( 𝐶 ≼* 𝐴 → 𝐶 ≼* 𝐴 ) |
2 |
|
endom |
⊢ ( 𝐴 ≈ 𝐵 → 𝐴 ≼ 𝐵 ) |
3 |
|
domwdom |
⊢ ( 𝐴 ≼ 𝐵 → 𝐴 ≼* 𝐵 ) |
4 |
2 3
|
syl |
⊢ ( 𝐴 ≈ 𝐵 → 𝐴 ≼* 𝐵 ) |
5 |
|
wdomtr |
⊢ ( ( 𝐶 ≼* 𝐴 ∧ 𝐴 ≼* 𝐵 ) → 𝐶 ≼* 𝐵 ) |
6 |
1 4 5
|
syl2anr |
⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐶 ≼* 𝐴 ) → 𝐶 ≼* 𝐵 ) |
7 |
|
id |
⊢ ( 𝐶 ≼* 𝐵 → 𝐶 ≼* 𝐵 ) |
8 |
|
ensym |
⊢ ( 𝐴 ≈ 𝐵 → 𝐵 ≈ 𝐴 ) |
9 |
|
endom |
⊢ ( 𝐵 ≈ 𝐴 → 𝐵 ≼ 𝐴 ) |
10 |
|
domwdom |
⊢ ( 𝐵 ≼ 𝐴 → 𝐵 ≼* 𝐴 ) |
11 |
8 9 10
|
3syl |
⊢ ( 𝐴 ≈ 𝐵 → 𝐵 ≼* 𝐴 ) |
12 |
|
wdomtr |
⊢ ( ( 𝐶 ≼* 𝐵 ∧ 𝐵 ≼* 𝐴 ) → 𝐶 ≼* 𝐴 ) |
13 |
7 11 12
|
syl2anr |
⊢ ( ( 𝐴 ≈ 𝐵 ∧ 𝐶 ≼* 𝐵 ) → 𝐶 ≼* 𝐴 ) |
14 |
6 13
|
impbida |
⊢ ( 𝐴 ≈ 𝐵 → ( 𝐶 ≼* 𝐴 ↔ 𝐶 ≼* 𝐵 ) ) |