Theorem iunssd

Description: Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 8-Apr-2021)

	Ref	Expression
Hypothesis	iunssd.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ⊆ 𝐶 )
Assertion	iunssd	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		iunssd.1	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ⊆ 𝐶 )
2	1	ralrimiva	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )
3		iunss	⊢ ( ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 ↔ ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )
4	2 3	sylibr	⊢ ( 𝜑 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 )