Theorem ssiun

Description: Subset implication for an indexed union. (Contributed by NM, 3-Sep-2003) (Proof shortened by Andrew Salmon, 25-Jul-2011)

	Ref	Expression
Assertion	ssiun	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		ssel	⊢ ( 𝐶 ⊆ 𝐵 → ( 𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵 ) )
2	1	reximi	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → ∃ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵 ) )
3		r19.37v	⊢ ( ∃ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐶 → 𝑦 ∈ 𝐵 ) → ( 𝑦 ∈ 𝐶 → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) )
4	2 3	syl	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → ( 𝑦 ∈ 𝐶 → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 ) )
5		eliun	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 )
6	4 5	imbitrrdi	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → ( 𝑦 ∈ 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ) )
7	6	ssrdv	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐶 ⊆ 𝐵 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐵 )