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Metamath Proof Explorer

Theorem meetdm

Description: Domain of meet function for a poset-type structure. (Contributed by NM, 16-Sep-2018)

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

Proof

Step Hyp Ref Expression
1 meetfval.u 
 |-  G = ( glb ` K )
2 meetfval.m 
 |-  ./\ = ( meet ` K )
3 1 2 meetfval2 
 |-  ( K e. V -> ./\ = { <. <. x , y >. , z >. | ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) } )
4 3 dmeqd 
 |-  ( K e. V -> dom ./\ = dom { <. <. x , y >. , z >. | ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) } )
5 dmoprab 
 |-  dom { <. <. x , y >. , z >. | ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) } = { <. x , y >. | E. z ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) }
6 fvex 
 |-  ( G ` { x , y } ) e. _V
7 6 isseti 
 |-  E. z z = ( G ` { x , y } )
8 19.42v 
 |-  ( E. z ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) <-> ( { x , y } e. dom G /\ E. z z = ( G ` { x , y } ) ) )
9 7 8 mpbiran2 
 |-  ( E. z ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) <-> { x , y } e. dom G )
10 9 opabbii 
 |-  { <. x , y >. | E. z ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) } = { <. x , y >. | { x , y } e. dom G }
11 5 10 eqtri 
 |-  dom { <. <. x , y >. , z >. | ( { x , y } e. dom G /\ z = ( G ` { x , y } ) ) } = { <. x , y >. | { x , y } e. dom G }
12 4 11 eqtrdi 
 |-  ( K e. V -> dom ./\ = { <. x , y >. | { x , y } e. dom G } )