Theorem acsficl

Description: A closure in an algebraic closure system is the union of the closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015)

	Ref	Expression
Hypothesis	acsdrscl.f	⊢ 𝐹 = ( mrCls ‘ 𝐶 )
Assertion	acsficl	⊢ ( ( 𝐶 ∈ ( ACS ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → ( 𝐹 ‘ 𝑆 ) = ∪ ( 𝐹 “ ( 𝒫 𝑆 ∩ Fin ) ) )

Proof

Step	Hyp	Ref	Expression
1		acsdrscl.f	⊢ 𝐹 = ( mrCls ‘ 𝐶 )
2		fveq2	⊢ ( 𝑠 = 𝑆 → ( 𝐹 ‘ 𝑠 ) = ( 𝐹 ‘ 𝑆 ) )
3		pweq	⊢ ( 𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆 )
4	3	ineq1d	⊢ ( 𝑠 = 𝑆 → ( 𝒫 𝑠 ∩ Fin ) = ( 𝒫 𝑆 ∩ Fin ) )
5	4	imaeq2d	⊢ ( 𝑠 = 𝑆 → ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) = ( 𝐹 “ ( 𝒫 𝑆 ∩ Fin ) ) )
6	5	unieqd	⊢ ( 𝑠 = 𝑆 → ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) = ∪ ( 𝐹 “ ( 𝒫 𝑆 ∩ Fin ) ) )
7	2 6	eqeq12d	⊢ ( 𝑠 = 𝑆 → ( ( 𝐹 ‘ 𝑠 ) = ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ↔ ( 𝐹 ‘ 𝑆 ) = ∪ ( 𝐹 “ ( 𝒫 𝑆 ∩ Fin ) ) ) )
8		isacs3lem	⊢ ( 𝐶 ∈ ( ACS ‘ 𝑋 ) → ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝐶 ) ) )
9	1	isacs4lem	⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝐶 ( ( toInc ‘ 𝑠 ) ∈ Dirset → ∪ 𝑠 ∈ 𝐶 ) ) → ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑡 ∈ 𝒫 𝒫 𝑋 ( ( toInc ‘ 𝑡 ) ∈ Dirset → ( 𝐹 ‘ ∪ 𝑡 ) = ∪ ( 𝐹 “ 𝑡 ) ) ) )
10	1	isacs5lem	⊢ ( ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑡 ∈ 𝒫 𝒫 𝑋 ( ( toInc ‘ 𝑡 ) ∈ Dirset → ( 𝐹 ‘ ∪ 𝑡 ) = ∪ ( 𝐹 “ 𝑡 ) ) ) → ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝐹 ‘ 𝑠 ) = ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ) )
11	8 9 10	3syl	⊢ ( 𝐶 ∈ ( ACS ‘ 𝑋 ) → ( 𝐶 ∈ ( Moore ‘ 𝑋 ) ∧ ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝐹 ‘ 𝑠 ) = ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) ) )
12	11	simprd	⊢ ( 𝐶 ∈ ( ACS ‘ 𝑋 ) → ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝐹 ‘ 𝑠 ) = ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) )
13	12	adantr	⊢ ( ( 𝐶 ∈ ( ACS ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → ∀ 𝑠 ∈ 𝒫 𝑋 ( 𝐹 ‘ 𝑠 ) = ∪ ( 𝐹 “ ( 𝒫 𝑠 ∩ Fin ) ) )
14		elfvdm	⊢ ( 𝐶 ∈ ( ACS ‘ 𝑋 ) → 𝑋 ∈ dom ACS )
15		elpw2g	⊢ ( 𝑋 ∈ dom ACS → ( 𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋 ) )
16	14 15	syl	⊢ ( 𝐶 ∈ ( ACS ‘ 𝑋 ) → ( 𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋 ) )
17	16	biimpar	⊢ ( ( 𝐶 ∈ ( ACS ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → 𝑆 ∈ 𝒫 𝑋 )
18	7 13 17	rspcdva	⊢ ( ( 𝐶 ∈ ( ACS ‘ 𝑋 ) ∧ 𝑆 ⊆ 𝑋 ) → ( 𝐹 ‘ 𝑆 ) = ∪ ( 𝐹 “ ( 𝒫 𝑆 ∩ Fin ) ) )