Theorem reliin

Description: An indexed intersection is a relation if at least one of the member of the indexed family is a relation. (Contributed by NM, 8-Mar-2014)

	Ref	Expression
Assertion	reliin	⊢ ( ∃ 𝑥 ∈ 𝐴 Rel 𝐵 → Rel ∩ 𝑥 ∈ 𝐴 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		iinss	⊢ ( ∃ 𝑥 ∈ 𝐴 𝐵 ⊆ ( V × V ) → ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ( V × V ) )
2		df-rel	⊢ ( Rel 𝐵 ↔ 𝐵 ⊆ ( V × V ) )
3	2	rexbii	⊢ ( ∃ 𝑥 ∈ 𝐴 Rel 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝐵 ⊆ ( V × V ) )
4		df-rel	⊢ ( Rel ∩ 𝑥 ∈ 𝐴 𝐵 ↔ ∩ 𝑥 ∈ 𝐴 𝐵 ⊆ ( V × V ) )
5	1 3 4	3imtr4i	⊢ ( ∃ 𝑥 ∈ 𝐴 Rel 𝐵 → Rel ∩ 𝑥 ∈ 𝐴 𝐵 )