Theorem mapfvd

Description: The value of a function that maps from B to A . (Contributed by AV, 2-Feb-2023)

	Ref	Expression
Hypotheses	mapfvd.m	⊢ 𝑀 = ( 𝐴 ↑m 𝐵 )
	mapfvd.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑀 )
	mapfvd.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
Assertion	mapfvd	⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) ∈ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		mapfvd.m	⊢ 𝑀 = ( 𝐴 ↑m 𝐵 )
2		mapfvd.f	⊢ ( 𝜑 → 𝐹 ∈ 𝑀 )
3		mapfvd.x	⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
4		elmapi	⊢ ( 𝐹 ∈ ( 𝐴 ↑m 𝐵 ) → 𝐹 : 𝐵 ⟶ 𝐴 )
5		ffvelrn	⊢ ( ( 𝐹 : 𝐵 ⟶ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐴 )
6	5	expcom	⊢ ( 𝑋 ∈ 𝐵 → ( 𝐹 : 𝐵 ⟶ 𝐴 → ( 𝐹 ‘ 𝑋 ) ∈ 𝐴 ) )
7	3 4 6	syl2imc	⊢ ( 𝐹 ∈ ( 𝐴 ↑m 𝐵 ) → ( 𝜑 → ( 𝐹 ‘ 𝑋 ) ∈ 𝐴 ) )
8	7 1	eleq2s	⊢ ( 𝐹 ∈ 𝑀 → ( 𝜑 → ( 𝐹 ‘ 𝑋 ) ∈ 𝐴 ) )
9	2 8	mpcom	⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) ∈ 𝐴 )