Theorem mptima2

Description: Image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021)

	Ref	Expression
Hypothesis	mptima2.1	⊢ ( 𝜑 → 𝐶 ⊆ 𝐴 )
Assertion	mptima2	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) “ 𝐶 ) = ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		mptima2.1	⊢ ( 𝜑 → 𝐶 ⊆ 𝐴 )
2		mptima	⊢ ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) “ 𝐶 ) = ran ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ↦ 𝐵 )
3		sseqin2	⊢ ( 𝐶 ⊆ 𝐴 ↔ ( 𝐴 ∩ 𝐶 ) = 𝐶 )
4	1 3	sylib	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐶 ) = 𝐶 )
5	4	mpteq1d	⊢ ( 𝜑 → ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ↦ 𝐵 ) = ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) )
6	5	rneqd	⊢ ( 𝜑 → ran ( 𝑥 ∈ ( 𝐴 ∩ 𝐶 ) ↦ 𝐵 ) = ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) )
7	2 6	syl5eq	⊢ ( 𝜑 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) “ 𝐶 ) = ran ( 𝑥 ∈ 𝐶 ↦ 𝐵 ) )