Theorem fneuni

Description: If B is finer than A , every element of A is a union of elements of B . (Contributed by Jeff Hankins, 11-Oct-2009)

	Ref	Expression
Assertion	fneuni	⊢ ( ( 𝐴 Fne 𝐵 ∧ 𝑆 ∈ 𝐴 ) → ∃ 𝑥 ( 𝑥 ⊆ 𝐵 ∧ 𝑆 = ∪ 𝑥 ) )

Proof

Step	Hyp	Ref	Expression
1		fnetg	⊢ ( 𝐴 Fne 𝐵 → 𝐴 ⊆ ( topGen ‘ 𝐵 ) )
2	1	sselda	⊢ ( ( 𝐴 Fne 𝐵 ∧ 𝑆 ∈ 𝐴 ) → 𝑆 ∈ ( topGen ‘ 𝐵 ) )
3		elfvdm	⊢ ( 𝑆 ∈ ( topGen ‘ 𝐵 ) → 𝐵 ∈ dom topGen )
4		eltg3	⊢ ( 𝐵 ∈ dom topGen → ( 𝑆 ∈ ( topGen ‘ 𝐵 ) ↔ ∃ 𝑥 ( 𝑥 ⊆ 𝐵 ∧ 𝑆 = ∪ 𝑥 ) ) )
5	3 4	syl	⊢ ( 𝑆 ∈ ( topGen ‘ 𝐵 ) → ( 𝑆 ∈ ( topGen ‘ 𝐵 ) ↔ ∃ 𝑥 ( 𝑥 ⊆ 𝐵 ∧ 𝑆 = ∪ 𝑥 ) ) )
6	5	ibi	⊢ ( 𝑆 ∈ ( topGen ‘ 𝐵 ) → ∃ 𝑥 ( 𝑥 ⊆ 𝐵 ∧ 𝑆 = ∪ 𝑥 ) )
7	2 6	syl	⊢ ( ( 𝐴 Fne 𝐵 ∧ 𝑆 ∈ 𝐴 ) → ∃ 𝑥 ( 𝑥 ⊆ 𝐵 ∧ 𝑆 = ∪ 𝑥 ) )