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 e. ( ACS ` X ) /\ S C_ X ) -> ( F ` S ) = U. ( F " ( ~P S i^i 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 i^i Fin ) = ( ~P S i^i Fin ) )
5	4	imaeq2d	|- ( s = S -> ( F " ( ~P s i^i Fin ) ) = ( F " ( ~P S i^i Fin ) ) )
6	5	unieqd	|- ( s = S -> U. ( F " ( ~P s i^i Fin ) ) = U. ( F " ( ~P S i^i Fin ) ) )
7	2 6	eqeq12d	|- ( s = S -> ( ( F ` s ) = U. ( F " ( ~P s i^i Fin ) ) <-> ( F ` S ) = U. ( F " ( ~P S i^i Fin ) ) ) )
8		isacs3lem	|- ( C e. ( ACS ` X ) -> ( C e. ( Moore ` X ) /\ A. s e. ~P C ( ( toInc ` s ) e. Dirset -> U. s e. C ) ) )
9	1	isacs4lem	|- ( ( C e. ( Moore ` X ) /\ A. s e. ~P C ( ( toInc ` s ) e. Dirset -> U. s e. C ) ) -> ( C e. ( Moore ` X ) /\ A. t e. ~P ~P X ( ( toInc ` t ) e. Dirset -> ( F ` U. t ) = U. ( F " t ) ) ) )
10	1	isacs5lem	|- ( ( C e. ( Moore ` X ) /\ A. t e. ~P ~P X ( ( toInc ` t ) e. Dirset -> ( F ` U. t ) = U. ( F " t ) ) ) -> ( C e. ( Moore ` X ) /\ A. s e. ~P X ( F ` s ) = U. ( F " ( ~P s i^i Fin ) ) ) )
11	8 9 10	3syl	|- ( C e. ( ACS ` X ) -> ( C e. ( Moore ` X ) /\ A. s e. ~P X ( F ` s ) = U. ( F " ( ~P s i^i Fin ) ) ) )
12	11	simprd	|- ( C e. ( ACS ` X ) -> A. s e. ~P X ( F ` s ) = U. ( F " ( ~P s i^i Fin ) ) )
13	12	adantr	|- ( ( C e. ( ACS ` X ) /\ S C_ X ) -> A. s e. ~P X ( F ` s ) = U. ( F " ( ~P s i^i Fin ) ) )
14		elfvdm	|- ( C e. ( ACS ` X ) -> X e. dom ACS )
15		elpw2g	|- ( X e. dom ACS -> ( S e. ~P X <-> S C_ X ) )
16	14 15	syl	|- ( C e. ( ACS ` X ) -> ( S e. ~P X <-> S C_ X ) )
17	16	biimpar	|- ( ( C e. ( ACS ` X ) /\ S C_ X ) -> S e. ~P X )
18	7 13 17	rspcdva	|- ( ( C e. ( ACS ` X ) /\ S C_ X ) -> ( F ` S ) = U. ( F " ( ~P S i^i Fin ) ) )