Theorem joincl

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

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

Proof

Step	Hyp	Ref	Expression
1		joincl.b	|- B = ( Base ` K )
2		joincl.j	|- .\/ = ( join ` K )
3		joincl.k	|- ( ph -> K e. V )
4		joincl.x	|- ( ph -> X e. B )
5		joincl.y	|- ( ph -> Y e. B )
6		joincl.e	|- ( ph -> <. X , Y >. e. dom .\/ )
7		eqid	|- ( lub ` K ) = ( lub ` K )
8	7 2 3 4 5	joinval	|- ( ph -> ( X .\/ Y ) = ( ( lub ` K ) ` { X , Y } ) )
9	7 2 3 4 5	joindef	|- ( ph -> ( <. X , Y >. e. dom .\/ <-> { X , Y } e. dom ( lub ` K ) ) )
10	6 9	mpbid	|- ( ph -> { X , Y } e. dom ( lub ` K ) )
11	1 7 3 10	lubcl	|- ( ph -> ( ( lub ` K ) ` { X , Y } ) e. B )
12	8 11	eqeltrd	|- ( ph -> ( X .\/ Y ) e. B )