Theorem rnmptssdff

Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 24-Jan-2025)

	Ref	Expression
Hypotheses	rnmptssdff.1	⊢ Ⅎ 𝑥 𝜑
	rnmptssdff.2	⊢ Ⅎ 𝑥 𝐴
	rnmptssdff.3	⊢ Ⅎ 𝑥 𝐶
	rnmptssdff.4	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
	rnmptssdff.5	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝐶 )
Assertion	rnmptssdff	⊢ ( 𝜑 → ran 𝐹 ⊆ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		rnmptssdff.1	⊢ Ⅎ 𝑥 𝜑
2		rnmptssdff.2	⊢ Ⅎ 𝑥 𝐴
3		rnmptssdff.3	⊢ Ⅎ 𝑥 𝐶
4		rnmptssdff.4	⊢ 𝐹 = ( 𝑥 ∈ 𝐴 ↦ 𝐵 )
5		rnmptssdff.5	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ 𝐶 )
6	1 5	ralrimia	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 )
7	2 3 4	rnmptssff	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶 )
8	6 7	syl	⊢ ( 𝜑 → ran 𝐹 ⊆ 𝐶 )