Step |
Hyp |
Ref |
Expression |
1 |
|
triun |
⊢ ( ∀ 𝑥 ∈ On Tr ( 𝑅1 ‘ 𝑥 ) → Tr ∪ 𝑥 ∈ On ( 𝑅1 ‘ 𝑥 ) ) |
2 |
|
r1tr |
⊢ Tr ( 𝑅1 ‘ 𝑥 ) |
3 |
2
|
a1i |
⊢ ( 𝑥 ∈ On → Tr ( 𝑅1 ‘ 𝑥 ) ) |
4 |
1 3
|
mprg |
⊢ Tr ∪ 𝑥 ∈ On ( 𝑅1 ‘ 𝑥 ) |
5 |
|
r1funlim |
⊢ ( Fun 𝑅1 ∧ Lim dom 𝑅1 ) |
6 |
5
|
simpli |
⊢ Fun 𝑅1 |
7 |
|
funiunfv |
⊢ ( Fun 𝑅1 → ∪ 𝑥 ∈ On ( 𝑅1 ‘ 𝑥 ) = ∪ ( 𝑅1 “ On ) ) |
8 |
6 7
|
ax-mp |
⊢ ∪ 𝑥 ∈ On ( 𝑅1 ‘ 𝑥 ) = ∪ ( 𝑅1 “ On ) |
9 |
|
treq |
⊢ ( ∪ 𝑥 ∈ On ( 𝑅1 ‘ 𝑥 ) = ∪ ( 𝑅1 “ On ) → ( Tr ∪ 𝑥 ∈ On ( 𝑅1 ‘ 𝑥 ) ↔ Tr ∪ ( 𝑅1 “ On ) ) ) |
10 |
8 9
|
ax-mp |
⊢ ( Tr ∪ 𝑥 ∈ On ( 𝑅1 ‘ 𝑥 ) ↔ Tr ∪ ( 𝑅1 “ On ) ) |
11 |
4 10
|
mpbi |
⊢ Tr ∪ ( 𝑅1 “ On ) |
12 |
|
trss |
⊢ ( Tr ∪ ( 𝑅1 “ On ) → ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → 𝐴 ⊆ ∪ ( 𝑅1 “ On ) ) ) |
13 |
11 12
|
ax-mp |
⊢ ( 𝐴 ∈ ∪ ( 𝑅1 “ On ) → 𝐴 ⊆ ∪ ( 𝑅1 “ On ) ) |