Step |
Hyp |
Ref |
Expression |
1 |
|
lautset.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
lautset.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
lautset.i |
⊢ 𝐼 = ( LAut ‘ 𝐾 ) |
4 |
1 2 3
|
islaut |
⊢ ( 𝐾 ∈ 𝑉 → ( 𝐹 ∈ 𝐼 ↔ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) ) ) |
5 |
4
|
simplbda |
⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) → ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) |
6 |
|
breq1 |
⊢ ( 𝑥 = 𝑋 → ( 𝑥 ≤ 𝑦 ↔ 𝑋 ≤ 𝑦 ) ) |
7 |
|
fveq2 |
⊢ ( 𝑥 = 𝑋 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑋 ) ) |
8 |
7
|
breq1d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) |
9 |
6 8
|
bibi12d |
⊢ ( 𝑥 = 𝑋 → ( ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝑋 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) ) |
10 |
|
breq2 |
⊢ ( 𝑦 = 𝑌 → ( 𝑋 ≤ 𝑦 ↔ 𝑋 ≤ 𝑌 ) ) |
11 |
|
fveq2 |
⊢ ( 𝑦 = 𝑌 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑌 ) ) |
12 |
11
|
breq2d |
⊢ ( 𝑦 = 𝑌 → ( ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) ) ) |
13 |
10 12
|
bibi12d |
⊢ ( 𝑦 = 𝑌 → ( ( 𝑋 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ↔ ( 𝑋 ≤ 𝑌 ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) ) ) ) |
14 |
9 13
|
rspc2v |
⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) ) ) ) |
15 |
5 14
|
mpan9 |
⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 ≤ 𝑌 ↔ ( 𝐹 ‘ 𝑋 ) ≤ ( 𝐹 ‘ 𝑌 ) ) ) |