Theorem lautcl

Description: A lattice automorphism value belongs to the base set. (Contributed by NM, 20-May-2012)

	Ref	Expression
Hypotheses	laut1o.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	laut1o.i	⊢ 𝐼 = ( LAut ‘ 𝐾 )
Assertion	lautcl	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 )

Proof

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		f1of	⊢ ( 𝐹 : 𝐵 –1-1-onto→ 𝐵 → 𝐹 : 𝐵 ⟶ 𝐵 )
5	3 4	syl	⊢ ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) → 𝐹 : 𝐵 ⟶ 𝐵 )
6	5	ffvelrnda	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼 ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 )