Theorem meetcl

Description: Closure of meet of elements in the domain. (Contributed by NM, 12-Sep-2018)

	Ref	Expression
Hypotheses	meetcl.b	|- B = ( Base ` K )
	meetcl.m	|- ./\ = ( meet ` K )
	meetcl.k	|- ( ph -> K e. V )
	meetcl.x	|- ( ph -> X e. B )
	meetcl.y	|- ( ph -> Y e. B )
	meetcl.e	|- ( ph -> <. X , Y >. e. dom ./\ )
Assertion	meetcl	|- ( ph -> ( X ./\ Y ) e. B )

Proof

Step	Hyp	Ref	Expression
1		meetcl.b	|- B = ( Base ` K )
2		meetcl.m	|- ./\ = ( meet ` K )
3		meetcl.k	|- ( ph -> K e. V )
4		meetcl.x	|- ( ph -> X e. B )
5		meetcl.y	|- ( ph -> Y e. B )
6		meetcl.e	|- ( ph -> <. X , Y >. e. dom ./\ )
7		eqid	|- ( glb ` K ) = ( glb ` K )
8	7 2 3 4 5	meetval	|- ( ph -> ( X ./\ Y ) = ( ( glb ` K ) ` { X , Y } ) )
9	7 2 3 4 5	meetdef	|- ( ph -> ( <. X , Y >. e. dom ./\ <-> { X , Y } e. dom ( glb ` K ) ) )
10	6 9	mpbid	|- ( ph -> { X , Y } e. dom ( glb ` K ) )
11	1 7 3 10	glbcl	|- ( ph -> ( ( glb ` K ) ` { X , Y } ) e. B )
12	8 11	eqeltrd	|- ( ph -> ( X ./\ Y ) e. B )