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Metamath Proof Explorer

Theorem setsmsbas

Description: The base set of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015) (Proof shortened by AV, 12-Nov-2024)

Ref Expression
Hypotheses setsms.x 
|- ( ph -> X = ( Base ` M ) )
setsms.d 
|- ( ph -> D = ( ( dist ` M ) |` ( X X. X ) ) )
setsms.k 
|- ( ph -> K = ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
Assertion setsmsbas 
|- ( ph -> X = ( Base ` K ) )

Proof

Step Hyp Ref Expression
1 setsms.x 
 |-  ( ph -> X = ( Base ` M ) )
2 setsms.d 
 |-  ( ph -> D = ( ( dist ` M ) |` ( X X. X ) ) )
3 setsms.k 
 |-  ( ph -> K = ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
4 baseid 
 |-  Base = Slot ( Base ` ndx )
5 tsetndxnbasendx 
 |-  ( TopSet ` ndx ) =/= ( Base ` ndx )
6 5 necomi 
 |-  ( Base ` ndx ) =/= ( TopSet ` ndx )
7 4 6 setsnid 
 |-  ( Base ` M ) = ( Base ` ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) )
8 3 fveq2d 
 |-  ( ph -> ( Base ` K ) = ( Base ` ( M sSet <. ( TopSet ` ndx ) , ( MetOpen ` D ) >. ) ) )
9 7 1 8 3eqtr4a 
 |-  ( ph -> X = ( Base ` K ) )