Step |
Hyp |
Ref |
Expression |
1 |
|
psrbag.d |
⊢ 𝐷 = { 𝑓 ∈ ( ℕ0 ↑m 𝐼 ) ∣ ( ◡ 𝑓 “ ℕ ) ∈ Fin } |
2 |
|
psrbagconf1o.s |
⊢ 𝑆 = { 𝑦 ∈ 𝐷 ∣ 𝑦 ∘r ≤ 𝐹 } |
3 |
|
simpl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → 𝐹 ∈ 𝐷 ) |
4 |
|
simpr |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → 𝑋 ∈ 𝑆 ) |
5 |
|
breq1 |
⊢ ( 𝑦 = 𝑋 → ( 𝑦 ∘r ≤ 𝐹 ↔ 𝑋 ∘r ≤ 𝐹 ) ) |
6 |
5 2
|
elrab2 |
⊢ ( 𝑋 ∈ 𝑆 ↔ ( 𝑋 ∈ 𝐷 ∧ 𝑋 ∘r ≤ 𝐹 ) ) |
7 |
4 6
|
sylib |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → ( 𝑋 ∈ 𝐷 ∧ 𝑋 ∘r ≤ 𝐹 ) ) |
8 |
7
|
simpld |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → 𝑋 ∈ 𝐷 ) |
9 |
1
|
psrbagf |
⊢ ( 𝑋 ∈ 𝐷 → 𝑋 : 𝐼 ⟶ ℕ0 ) |
10 |
8 9
|
syl |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → 𝑋 : 𝐼 ⟶ ℕ0 ) |
11 |
7
|
simprd |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → 𝑋 ∘r ≤ 𝐹 ) |
12 |
1
|
psrbagcon |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑋 : 𝐼 ⟶ ℕ0 ∧ 𝑋 ∘r ≤ 𝐹 ) → ( ( 𝐹 ∘f − 𝑋 ) ∈ 𝐷 ∧ ( 𝐹 ∘f − 𝑋 ) ∘r ≤ 𝐹 ) ) |
13 |
3 10 11 12
|
syl3anc |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → ( ( 𝐹 ∘f − 𝑋 ) ∈ 𝐷 ∧ ( 𝐹 ∘f − 𝑋 ) ∘r ≤ 𝐹 ) ) |
14 |
|
breq1 |
⊢ ( 𝑦 = ( 𝐹 ∘f − 𝑋 ) → ( 𝑦 ∘r ≤ 𝐹 ↔ ( 𝐹 ∘f − 𝑋 ) ∘r ≤ 𝐹 ) ) |
15 |
14 2
|
elrab2 |
⊢ ( ( 𝐹 ∘f − 𝑋 ) ∈ 𝑆 ↔ ( ( 𝐹 ∘f − 𝑋 ) ∈ 𝐷 ∧ ( 𝐹 ∘f − 𝑋 ) ∘r ≤ 𝐹 ) ) |
16 |
13 15
|
sylibr |
⊢ ( ( 𝐹 ∈ 𝐷 ∧ 𝑋 ∈ 𝑆 ) → ( 𝐹 ∘f − 𝑋 ) ∈ 𝑆 ) |