Theorem iunss1

Description: Subclass theorem for indexed union. (Contributed by NM, 10-Dec-2004) (Proof shortened by Andrew Salmon, 25-Jul-2011)

	Ref	Expression
Assertion	iunss1	⊢ ( 𝐴 ⊆ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ssrexv	⊢ ( 𝐴 ⊆ 𝐵 → ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 → ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 ) )
2		eliun	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 )
3		eliun	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃ 𝑥 ∈ 𝐵 𝑦 ∈ 𝐶 )
4	1 2 3	3imtr4g	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ) )
5	4	ssrdv	⊢ ( 𝐴 ⊆ 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑥 ∈ 𝐵 𝐶 )