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Metamath Proof Explorer

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 ffvelcdmda ⊢ ( ( 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐴 )