Theorem resmptd

Description: Restriction of the mapping operation, deduction form. (Contributed by Glauco Siliprandi, 11-Dec-2019)

	Ref	Expression
Hypothesis	resmptd.b	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
Assertion	resmptd	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ↾ 𝐵 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		resmptd.b	⊢ ( 𝜑 → 𝐵 ⊆ 𝐴 )
2		resmpt	⊢ ( 𝐵 ⊆ 𝐴 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ↾ 𝐵 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )
3	1 2	syl	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐶 ) ↾ 𝐵 ) = ( 𝑥 ∈ 𝐵 ↦ 𝐶 ) )