Step |
Hyp |
Ref |
Expression |
1 |
|
frege96d.r |
⊢ ( 𝜑 → 𝑅 ∈ V ) |
2 |
|
frege96d.a |
⊢ ( 𝜑 → 𝐴 ∈ V ) |
3 |
|
frege96d.b |
⊢ ( 𝜑 → 𝐵 ∈ V ) |
4 |
|
frege96d.c |
⊢ ( 𝜑 → 𝐶 ∈ V ) |
5 |
|
frege96d.ac |
⊢ ( 𝜑 → 𝐴 ( t+ ‘ 𝑅 ) 𝐶 ) |
6 |
|
frege96d.cb |
⊢ ( 𝜑 → 𝐶 𝑅 𝐵 ) |
7 |
|
brcogw |
⊢ ( ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V ) ∧ ( 𝐴 ( t+ ‘ 𝑅 ) 𝐶 ∧ 𝐶 𝑅 𝐵 ) ) → 𝐴 ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) 𝐵 ) |
8 |
2 3 4 5 6 7
|
syl32anc |
⊢ ( 𝜑 → 𝐴 ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) 𝐵 ) |
9 |
|
trclfvlb |
⊢ ( 𝑅 ∈ V → 𝑅 ⊆ ( t+ ‘ 𝑅 ) ) |
10 |
|
coss1 |
⊢ ( 𝑅 ⊆ ( t+ ‘ 𝑅 ) → ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) ) |
11 |
1 9 10
|
3syl |
⊢ ( 𝜑 → ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) ) |
12 |
|
trclfvcotrg |
⊢ ( ( t+ ‘ 𝑅 ) ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( t+ ‘ 𝑅 ) |
13 |
11 12
|
sstrdi |
⊢ ( 𝜑 → ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) ⊆ ( t+ ‘ 𝑅 ) ) |
14 |
13
|
ssbrd |
⊢ ( 𝜑 → ( 𝐴 ( 𝑅 ∘ ( t+ ‘ 𝑅 ) ) 𝐵 → 𝐴 ( t+ ‘ 𝑅 ) 𝐵 ) ) |
15 |
8 14
|
mpd |
⊢ ( 𝜑 → 𝐴 ( t+ ‘ 𝑅 ) 𝐵 ) |