Step |
Hyp |
Ref |
Expression |
1 |
|
dfsmo2 |
⊢ ( Smo 𝐹 ↔ ( 𝐹 : dom 𝐹 ⟶ On ∧ Ord dom 𝐹 ∧ ∀ 𝑥 ∈ dom 𝐹 ∀ 𝑦 ∈ 𝑥 ( 𝐹 ‘ 𝑦 ) ∈ ( 𝐹 ‘ 𝑥 ) ) ) |
2 |
1
|
simp1bi |
⊢ ( Smo 𝐹 → 𝐹 : dom 𝐹 ⟶ On ) |
3 |
|
ffvelrn |
⊢ ( ( 𝐹 : dom 𝐹 ⟶ On ∧ 𝐵 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝐵 ) ∈ On ) |
4 |
3
|
expcom |
⊢ ( 𝐵 ∈ dom 𝐹 → ( 𝐹 : dom 𝐹 ⟶ On → ( 𝐹 ‘ 𝐵 ) ∈ On ) ) |
5 |
2 4
|
syl5 |
⊢ ( 𝐵 ∈ dom 𝐹 → ( Smo 𝐹 → ( 𝐹 ‘ 𝐵 ) ∈ On ) ) |
6 |
|
ndmfv |
⊢ ( ¬ 𝐵 ∈ dom 𝐹 → ( 𝐹 ‘ 𝐵 ) = ∅ ) |
7 |
|
0elon |
⊢ ∅ ∈ On |
8 |
6 7
|
eqeltrdi |
⊢ ( ¬ 𝐵 ∈ dom 𝐹 → ( 𝐹 ‘ 𝐵 ) ∈ On ) |
9 |
8
|
a1d |
⊢ ( ¬ 𝐵 ∈ dom 𝐹 → ( Smo 𝐹 → ( 𝐹 ‘ 𝐵 ) ∈ On ) ) |
10 |
5 9
|
pm2.61i |
⊢ ( Smo 𝐹 → ( 𝐹 ‘ 𝐵 ) ∈ On ) |