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Metamath Proof Explorer

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 } ) ) } )