Step |
Hyp |
Ref |
Expression |
1 |
|
frege109d.r |
⊢ ( 𝜑 → 𝑅 ∈ V ) |
2 |
|
frege109d.a |
⊢ ( 𝜑 → 𝐴 = ( 𝑈 ∪ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) ) |
3 |
|
trclfvlb |
⊢ ( 𝑅 ∈ V → 𝑅 ⊆ ( t+ ‘ 𝑅 ) ) |
4 |
|
imass1 |
⊢ ( 𝑅 ⊆ ( t+ ‘ 𝑅 ) → ( 𝑅 “ 𝑈 ) ⊆ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) |
5 |
1 3 4
|
3syl |
⊢ ( 𝜑 → ( 𝑅 “ 𝑈 ) ⊆ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) |
6 |
|
coss1 |
⊢ ( 𝑅 ⊆ ( t+ ‘ 𝑅 ) → ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) ) |
7 |
1 3 6
|
3syl |
⊢ ( 𝜑 → ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) ) |
8 |
|
trclfvcotrg |
⊢ ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( t+ ‘ 𝑅 ) |
9 |
7 8
|
sstrdi |
⊢ ( 𝜑 → ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( t+ ‘ 𝑅 ) ) |
10 |
|
imass1 |
⊢ ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( t+ ‘ 𝑅 ) → ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 ) ⊆ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) |
11 |
9 10
|
syl |
⊢ ( 𝜑 → ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 ) ⊆ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) |
12 |
5 11
|
unssd |
⊢ ( 𝜑 → ( ( 𝑅 “ 𝑈 ) ∪ ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 ) ) ⊆ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) |
13 |
|
ssun2 |
⊢ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ⊆ ( 𝑈 ∪ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) |
14 |
12 13
|
sstrdi |
⊢ ( 𝜑 → ( ( 𝑅 “ 𝑈 ) ∪ ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 ) ) ⊆ ( 𝑈 ∪ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) ) |
15 |
2
|
imaeq2d |
⊢ ( 𝜑 → ( 𝑅 “ 𝐴 ) = ( 𝑅 “ ( 𝑈 ∪ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) ) ) |
16 |
|
imaundi |
⊢ ( 𝑅 “ ( 𝑈 ∪ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) ) = ( ( 𝑅 “ 𝑈 ) ∪ ( 𝑅 “ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) ) |
17 |
|
imaco |
⊢ ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 ) = ( 𝑅 “ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) |
18 |
17
|
eqcomi |
⊢ ( 𝑅 “ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) = ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 ) |
19 |
18
|
uneq2i |
⊢ ( ( 𝑅 “ 𝑈 ) ∪ ( 𝑅 “ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) ) = ( ( 𝑅 “ 𝑈 ) ∪ ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 ) ) |
20 |
16 19
|
eqtri |
⊢ ( 𝑅 “ ( 𝑈 ∪ ( ( t+ ‘ 𝑅 ) “ 𝑈 ) ) ) = ( ( 𝑅 “ 𝑈 ) ∪ ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 ) ) |
21 |
15 20
|
eqtrdi |
⊢ ( 𝜑 → ( 𝑅 “ 𝐴 ) = ( ( 𝑅 “ 𝑈 ) ∪ ( ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) “ 𝑈 ) ) ) |
22 |
14 21 2
|
3sstr4d |
⊢ ( 𝜑 → ( 𝑅 “ 𝐴 ) ⊆ 𝐴 ) |