Theorem iunss2

Description: A subclass condition on the members of two indexed classes C ( x ) and D ( y ) that implies a subclass relation on their indexed unions. Generalization of Proposition 8.6 of TakeutiZaring p. 59. Compare uniss2 . (Contributed by NM, 9-Dec-2004)

	Ref	Expression
Assertion	iunss2	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷 )

Proof

Step	Hyp	Ref	Expression
1		ssiun	⊢ ( ∃ 𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷 )
2	1	ralimi	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → ∀ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷 )
3		iunss	⊢ ( ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷 ↔ ∀ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷 )
4	2 3	sylibr	⊢ ( ∀ 𝑥 ∈ 𝐴 ∃ 𝑦 ∈ 𝐵 𝐶 ⊆ 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ 𝐵 𝐷 )