Theorem iunssdf

Description: Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 24-Jan-2025)

	Ref	Expression
Hypotheses	iunssdf.1	⊢ Ⅎ 𝑥 𝜑
	iunssdf.2	⊢ Ⅎ 𝑥 𝐶
	iunssdf.3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ⊆ 𝐶 )
Assertion	iunssdf	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		iunssdf.1	⊢ Ⅎ 𝑥 𝜑
2		iunssdf.2	⊢ Ⅎ 𝑥 𝐶
3		iunssdf.3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ⊆ 𝐶 )
4	1 3	ralrimia	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )
5	2	iunssf	⊢ ( ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )
6	4 5	sylibr	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )