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Metamath Proof Explorer

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 )