Step |
Hyp |
Ref |
Expression |
1 |
|
relcnv |
⊢ Rel ◡ 𝐴 |
2 |
|
ctex |
⊢ ( 𝐴 ≼ ω → 𝐴 ∈ V ) |
3 |
|
cnvexg |
⊢ ( 𝐴 ∈ V → ◡ 𝐴 ∈ V ) |
4 |
2 3
|
syl |
⊢ ( 𝐴 ≼ ω → ◡ 𝐴 ∈ V ) |
5 |
|
cnven |
⊢ ( ( Rel ◡ 𝐴 ∧ ◡ 𝐴 ∈ V ) → ◡ 𝐴 ≈ ◡ ◡ 𝐴 ) |
6 |
1 4 5
|
sylancr |
⊢ ( 𝐴 ≼ ω → ◡ 𝐴 ≈ ◡ ◡ 𝐴 ) |
7 |
|
cnvcnvss |
⊢ ◡ ◡ 𝐴 ⊆ 𝐴 |
8 |
|
ssdomg |
⊢ ( 𝐴 ∈ V → ( ◡ ◡ 𝐴 ⊆ 𝐴 → ◡ ◡ 𝐴 ≼ 𝐴 ) ) |
9 |
2 7 8
|
mpisyl |
⊢ ( 𝐴 ≼ ω → ◡ ◡ 𝐴 ≼ 𝐴 ) |
10 |
|
endomtr |
⊢ ( ( ◡ 𝐴 ≈ ◡ ◡ 𝐴 ∧ ◡ ◡ 𝐴 ≼ 𝐴 ) → ◡ 𝐴 ≼ 𝐴 ) |
11 |
6 9 10
|
syl2anc |
⊢ ( 𝐴 ≼ ω → ◡ 𝐴 ≼ 𝐴 ) |
12 |
|
domtr |
⊢ ( ( ◡ 𝐴 ≼ 𝐴 ∧ 𝐴 ≼ ω ) → ◡ 𝐴 ≼ ω ) |
13 |
11 12
|
mpancom |
⊢ ( 𝐴 ≼ ω → ◡ 𝐴 ≼ ω ) |