Step |
Hyp |
Ref |
Expression |
1 |
|
simpl1 |
⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐹 : 𝐴 ⟶ 𝐵 ) |
2 |
1
|
ffnd |
⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → 𝐹 Fn 𝐴 ) |
3 |
|
simpl2 |
⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → Smo 𝐹 ) |
4 |
|
smodm2 |
⊢ ( ( 𝐹 Fn 𝐴 ∧ Smo 𝐹 ) → Ord 𝐴 ) |
5 |
2 3 4
|
syl2anc |
⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → Ord 𝐴 ) |
6 |
|
ordelord |
⊢ ( ( Ord 𝐴 ∧ 𝑥 ∈ 𝐴 ) → Ord 𝑥 ) |
7 |
5 6
|
sylancom |
⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → Ord 𝑥 ) |
8 |
|
simpl3 |
⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → Ord 𝐵 ) |
9 |
|
simpr |
⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) |
10 |
|
smogt |
⊢ ( ( 𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ) |
11 |
2 3 9 10
|
syl3anc |
⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ) |
12 |
|
ffvelrn |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) |
13 |
12
|
3ad2antl1 |
⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) |
14 |
|
ordtr2 |
⊢ ( ( Ord 𝑥 ∧ Ord 𝐵 ) → ( ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) → 𝑥 ∈ 𝐵 ) ) |
15 |
14
|
imp |
⊢ ( ( ( Ord 𝑥 ∧ Ord 𝐵 ) ∧ ( 𝑥 ⊆ ( 𝐹 ‘ 𝑥 ) ∧ ( 𝐹 ‘ 𝑥 ) ∈ 𝐵 ) ) → 𝑥 ∈ 𝐵 ) |
16 |
7 8 11 13 15
|
syl22anc |
⊢ ( ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐵 ) |
17 |
16
|
ex |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵 ) ) |
18 |
17
|
ssrdv |
⊢ ( ( 𝐹 : 𝐴 ⟶ 𝐵 ∧ Smo 𝐹 ∧ Ord 𝐵 ) → 𝐴 ⊆ 𝐵 ) |