Theorem mrelatlubALT

Description: Least upper bounds in a Moore space are realized by the closure of the union. (Contributed by Stefan O'Rear, 31-Jan-2015) (Proof shortened by Zhi Wang, 29-Sep-2024) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	mreclatGOOD.i	⊢ 𝐼 = ( toInc ‘ 𝐶 )
	mrelatlubALT.f	⊢ 𝐹 = ( mrCls ‘ 𝐶 )
	mrelatlubALT.l	⊢ 𝐿 = ( lub ‘ 𝐼 )
Assertion	mrelatlubALT	⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝐶 ) → ( 𝐿 ‘ 𝑈 ) = ( 𝐹 ‘ ∪ 𝑈 ) )

Proof

Step	Hyp	Ref	Expression
1		mreclatGOOD.i	⊢ 𝐼 = ( toInc ‘ 𝐶 )
2		mrelatlubALT.f	⊢ 𝐹 = ( mrCls ‘ 𝐶 )
3		mrelatlubALT.l	⊢ 𝐿 = ( lub ‘ 𝐼 )
4		simpl	⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝐶 ) → 𝐶 ∈ ( Moore ‘ 𝑋 ) )
5		simpr	⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝐶 ) → 𝑈 ⊆ 𝐶 )
6	3	a1i	⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝐶 ) → 𝐿 = ( lub ‘ 𝐼 ) )
7		mreuniss	⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝐶 ) → ∪ 𝑈 ⊆ 𝑋 )
8	2	mrcval	⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∪ 𝑈 ⊆ 𝑋 ) → ( 𝐹 ‘ ∪ 𝑈 ) = ∩ { 𝑥 ∈ 𝐶 ∣ ∪ 𝑈 ⊆ 𝑥 } )
9	7 8	syldan	⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝐶 ) → ( 𝐹 ‘ ∪ 𝑈 ) = ∩ { 𝑥 ∈ 𝐶 ∣ ∪ 𝑈 ⊆ 𝑥 } )
10	2	mrccl	⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∪ 𝑈 ⊆ 𝑋 ) → ( 𝐹 ‘ ∪ 𝑈 ) ∈ 𝐶 )
11	7 10	syldan	⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝐶 ) → ( 𝐹 ‘ ∪ 𝑈 ) ∈ 𝐶 )
12	1 4 5 6 9 11	ipolub	⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ 𝑈 ⊆ 𝐶 ) → ( 𝐿 ‘ 𝑈 ) = ( 𝐹 ‘ ∪ 𝑈 ) )