Theorem reliun

Description: An indexed union is a relation iff each member of its indexed family is a relation. (Contributed by NM, 19-Dec-2008) (Proof shortened by SN, 2-Feb-2025)

	Ref	Expression
Assertion	reliun	⊢ ( Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 Rel 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		iunss	⊢ ( ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ( V × V ) ↔ ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ ( V × V ) )
2		df-rel	⊢ ( Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ ( V × V ) )
3		df-rel	⊢ ( Rel 𝐵 ↔ 𝐵 ⊆ ( V × V ) )
4	3	ralbii	⊢ ( ∀ 𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 𝐵 ⊆ ( V × V ) )
5	1 2 4	3bitr4i	⊢ ( Rel ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∀ 𝑥 ∈ 𝐴 Rel 𝐵 )