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Metamath Proof Explorer

Theorem islaut

Description: The predicate "is a lattice automorphism". (Contributed by NM, 11-May-2012)

Ref Expression
Hypotheses lautset.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
lautset.l ⊢ ≤ = ( le ‘ 𝐾 )
lautset.i ⊢ 𝐼 = ( LAut ‘ 𝐾 )
Assertion islaut ( 𝐾 ∈ 𝐴 → ( 𝐹 ∈ 𝐼 ↔ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) ) )

Proof

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 lautset ⊢ ( 𝐾 ∈ 𝐴 → 𝐼 = { 𝑓 ∣ ( 𝑓 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ≤ ( 𝑓 ‘ 𝑦 ) ) ) } )
5 4 eleq2d ⊢ ( 𝐾 ∈ 𝐴 → ( 𝐹 ∈ 𝐼 ↔ 𝐹 ∈ { 𝑓 ∣ ( 𝑓 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ≤ ( 𝑓 ‘ 𝑦 ) ) ) } ) )
6 f1of ⊢ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 → 𝐹 : 𝐵 ⟶ 𝐵 )
7 1 fvexi ⊢ 𝐵 ∈ V
8 fex ⊢ ( ( 𝐹 : 𝐵 ⟶ 𝐵 ∧ 𝐵 ∈ V ) → 𝐹 ∈ V )
9 6 7 8 sylancl ⊢ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 → 𝐹 ∈ V )
10 9 adantr ⊢ ( ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) → 𝐹 ∈ V )
11 f1oeq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 : 𝐵 –1-1-onto→ 𝐵 ↔ 𝐹 : 𝐵 –1-1-onto→ 𝐵 ) )
12 fveq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) )
13 fveq1 ⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
14 12 13 breq12d ⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑥 ) ≤ ( 𝑓 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) )
15 14 bibi2d ⊢ ( 𝑓 = 𝐹 → ( ( 𝑥 ≤ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ≤ ( 𝑓 ‘ 𝑦 ) ) ↔ ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) )
16 15 2ralbidv ⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ≤ ( 𝑓 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) )
17 11 16 anbi12d ⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ≤ ( 𝑓 ‘ 𝑦 ) ) ) ↔ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) ) )
18 10 17 elab3 ⊢ ( 𝐹 ∈ { 𝑓 ∣ ( 𝑓 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ≤ ( 𝑓 ‘ 𝑦 ) ) ) } ↔ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) )
19 5 18 bitrdi ⊢ ( 𝐾 ∈ 𝐴 → ( 𝐹 ∈ 𝐼 ↔ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 ∀ 𝑦 ∈ 𝐵 ( 𝑥 ≤ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ≤ ( 𝐹 ‘ 𝑦 ) ) ) ) )