Theorem symgfv

Description: The function value of a permutation. (Contributed by AV, 1-Jan-2019)

	Ref	Expression
Hypotheses	symgbas.1	⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
	symgbas.2	⊢ 𝐵 = ( Base ‘ 𝐺 )
Assertion	symgfv	⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		symgbas.1	⊢ 𝐺 = ( SymGrp ‘ 𝐴 )
2		symgbas.2	⊢ 𝐵 = ( Base ‘ 𝐺 )
3	1 2	symgbasf	⊢ ( 𝐹 ∈ 𝐵 → 𝐹 : 𝐴 ⟶ 𝐴 )
4	3	ffvelrnda	⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐴 )