Theorem meetfval2

Description: Value of meet function for a poset. (Contributed by NM, 12-Sep-2011) (Revised by NM, 9-Sep-2018)

	Ref	Expression
Hypotheses	meetfval.u	|- G = ( glb ` K )
	meetfval.m	|- ./\ = ( meet ` K )
Assertion	meetfval2	|- ( K e. V -> ./\ = { <. <. x , y >. , z >. | ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) } )

Proof

Step	Hyp	Ref	Expression
1		meetfval.u	|- G = ( glb ` K )
2		meetfval.m	|- ./\ = ( meet ` K )
3	1 2	meetfval	|- ( K e. V -> ./\ = { <. <. x , y >. , z >. | { x , y } G z } )
4	1	glbfun	|- Fun G
5		funbrfv2b	|- ( Fun G -> ( { x , y } G z <-> ( { x , y } e. dom G /\ ( G ` { x , y } ) = z ) ) )
6	4 5	ax-mp	|- ( { x , y } G z <-> ( { x , y } e. dom G /\ ( G ` { x , y } ) = z ) )
7		eqcom	|- ( ( G ` { x , y } ) = z <-> z = ( G ` { x , y } ) )
8	7	anbi2i	|- ( ( { x , y } e. dom G /\ ( G ` { x , y } ) = z ) <-> ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) )
9	6 8	bitri	|- ( { x , y } G z <-> ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) )
10	9	oprabbii	|- { <. <. x , y >. , z >. | { x , y } G z } = { <. <. x , y >. , z >. | ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) }
11	3 10	eqtrdi	|- ( K e. V -> ./\ = { <. <. x , y >. , z >. | ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) } )