Description: One-to-one property of a lattice automorphism. (Contributed by NM, 20-May-2012)
Ref | Expression | ||
---|---|---|---|
Hypotheses | laut1o.b | ⊢ 𝐵 = ( Base ‘ 𝐾 ) | |
laut1o.i | ⊢ 𝐼 = ( LAut ‘ 𝐾 ) | ||
Assertion | laut11 | ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | laut1o.b | ⊢ 𝐵 = ( Base ‘ 𝐾 ) | |
2 | laut1o.i | ⊢ 𝐼 = ( LAut ‘ 𝐾 ) | |
3 | 1 2 | laut1o | ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐵 ) |
4 | f1of1 | ⊢ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 → 𝐹 : 𝐵 –1-1→ 𝐵 ) | |
5 | 3 4 | syl | ⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) → 𝐹 : 𝐵 –1-1→ 𝐵 ) |
6 | f1fveq | ⊢ ( ( 𝐹 : 𝐵 –1-1→ 𝐵 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) ) | |
7 | 5 6 | sylan | ⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( ( 𝐹 ‘ 𝑋 ) = ( 𝐹 ‘ 𝑌 ) ↔ 𝑋 = 𝑌 ) ) |