Theorem elimampt

Description: Membership in the image of a mapping. (Contributed by Thierry Arnoux, 3-Jan-2022)

	Ref	Expression
Hypotheses	elimampt.f	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
	elimampt.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑊 )
	elimampt.d	⊢ ( 𝜑 → 𝐷 ⊆ 𝐴 )
Assertion	elimampt	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐹 “ 𝐷 ) ↔ ∃ 𝑥 ∈ 𝐷 𝐶 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		elimampt.f	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
2		elimampt.c	⊢ ( 𝜑 → 𝐶 ∈ 𝑊 )
3		elimampt.d	⊢ ( 𝜑 → 𝐷 ⊆ 𝐴 )
4		df-ima	⊢ ( 𝐹 “ 𝐷 ) = ran ( 𝐹 ↾ 𝐷 )
5	4	eleq2i	⊢ ( 𝐶 ∈ ( 𝐹 “ 𝐷 ) ↔ 𝐶 ∈ ran ( 𝐹 ↾ 𝐷 ) )
6	1	reseq1i	⊢ ( 𝐹 ↾ 𝐷 ) = ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝐷 )
7		resmpt	⊢ ( 𝐷 ⊆ 𝐴 → ( ( 𝑥 ∈ 𝐴 ↦ 𝐵 ) ↾ 𝐷 ) = ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) )
8	6 7	eqtrid	⊢ ( 𝐷 ⊆ 𝐴 → ( 𝐹 ↾ 𝐷 ) = ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) )
9	8	rneqd	⊢ ( 𝐷 ⊆ 𝐴 → ran ( 𝐹 ↾ 𝐷 ) = ran ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) )
10	9	eleq2d	⊢ ( 𝐷 ⊆ 𝐴 → ( 𝐶 ∈ ran ( 𝐹 ↾ 𝐷 ) ↔ 𝐶 ∈ ran ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) ) )
11	3 10	syl	⊢ ( 𝜑 → ( 𝐶 ∈ ran ( 𝐹 ↾ 𝐷 ) ↔ 𝐶 ∈ ran ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) ) )
12		eqid	⊢ ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) = ( 𝑥 ∈ 𝐷 ↦ 𝐵 )
13	12	elrnmpt	⊢ ( 𝐶 ∈ 𝑊 → ( 𝐶 ∈ ran ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐷 𝐶 = 𝐵 ) )
14	2 13	syl	⊢ ( 𝜑 → ( 𝐶 ∈ ran ( 𝑥 ∈ 𝐷 ↦ 𝐵 ) ↔ ∃ 𝑥 ∈ 𝐷 𝐶 = 𝐵 ) )
15	11 14	bitrd	⊢ ( 𝜑 → ( 𝐶 ∈ ran ( 𝐹 ↾ 𝐷 ) ↔ ∃ 𝑥 ∈ 𝐷 𝐶 = 𝐵 ) )
16	5 15	bitrid	⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐹 “ 𝐷 ) ↔ ∃ 𝑥 ∈ 𝐷 𝐶 = 𝐵 ) )