Theorem ss2iun

Description: Subclass theorem for indexed union. (Contributed by NM, 26-Nov-2003) (Proof shortened by Andrew Salmon, 25-Jul-2011)

	Ref	Expression
Assertion	ss2iun	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		ssel	⊢ ( 𝐵 ⊆ 𝐶 → ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
2	1	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) )
3		rexim	⊢ ( ∀ 𝑥 ∈ 𝐴 ( 𝑦 ∈ 𝐵 → 𝑦 ∈ 𝐶 ) → ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ) )
4	2 3	syl	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ( ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 → ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 ) )
5		eliun	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 )
6		eliun	⊢ ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃ 𝑥 ∈ 𝐴 𝑦 ∈ 𝐶 )
7	4 5 6	3imtr4g	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ( 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 → 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ) )
8	7	ssrdv	⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶 )