Step |
Hyp |
Ref |
Expression |
1 |
|
r1funlim |
⊢ ( Fun 𝑅1 ∧ Lim dom 𝑅1 ) |
2 |
1
|
simpri |
⊢ Lim dom 𝑅1 |
3 |
|
limord |
⊢ ( Lim dom 𝑅1 → Ord dom 𝑅1 ) |
4 |
|
ordtr1 |
⊢ ( Ord dom 𝑅1 → ( ( 𝑥 ∈ 𝐴 ∧ 𝐴 ∈ dom 𝑅1 ) → 𝑥 ∈ dom 𝑅1 ) ) |
5 |
2 3 4
|
mp2b |
⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐴 ∈ dom 𝑅1 ) → 𝑥 ∈ dom 𝑅1 ) |
6 |
5
|
ancoms |
⊢ ( ( 𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ dom 𝑅1 ) |
7 |
|
rankonidlem |
⊢ ( 𝑥 ∈ dom 𝑅1 → ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑥 ) = 𝑥 ) ) |
8 |
6 7
|
syl |
⊢ ( ( 𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∧ ( rank ‘ 𝑥 ) = 𝑥 ) ) |
9 |
8
|
simprd |
⊢ ( ( 𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴 ) → ( rank ‘ 𝑥 ) = 𝑥 ) |
10 |
|
simpr |
⊢ ( ( 𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) |
11 |
9 10
|
eqeltrd |
⊢ ( ( 𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴 ) → ( rank ‘ 𝑥 ) ∈ 𝐴 ) |
12 |
8
|
simpld |
⊢ ( ( 𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ∪ ( 𝑅1 “ On ) ) |
13 |
|
simpl |
⊢ ( ( 𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴 ) → 𝐴 ∈ dom 𝑅1 ) |
14 |
|
rankr1ag |
⊢ ( ( 𝑥 ∈ ∪ ( 𝑅1 “ On ) ∧ 𝐴 ∈ dom 𝑅1 ) → ( 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ↔ ( rank ‘ 𝑥 ) ∈ 𝐴 ) ) |
15 |
12 13 14
|
syl2anc |
⊢ ( ( 𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴 ) → ( 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ↔ ( rank ‘ 𝑥 ) ∈ 𝐴 ) ) |
16 |
11 15
|
mpbird |
⊢ ( ( 𝐴 ∈ dom 𝑅1 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) |
17 |
16
|
ex |
⊢ ( 𝐴 ∈ dom 𝑅1 → ( 𝑥 ∈ 𝐴 → 𝑥 ∈ ( 𝑅1 ‘ 𝐴 ) ) ) |
18 |
17
|
ssrdv |
⊢ ( 𝐴 ∈ dom 𝑅1 → 𝐴 ⊆ ( 𝑅1 ‘ 𝐴 ) ) |