Metamath Proof Explorer


Theorem rngbase

Description: The base set of a constructed ring. (Contributed by Mario Carneiro, 2-Oct-2013) (Revised by Mario Carneiro, 30-Apr-2015)

Ref Expression
Hypothesis rngfn.r
|- R = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. }
Assertion rngbase
|- ( B e. V -> B = ( Base ` R ) )

Proof

Step Hyp Ref Expression
1 rngfn.r
 |-  R = { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. }
2 1 rngstr
 |-  R Struct <. 1 , 3 >.
3 baseid
 |-  Base = Slot ( Base ` ndx )
4 snsstp1
 |-  { <. ( Base ` ndx ) , B >. } C_ { <. ( Base ` ndx ) , B >. , <. ( +g ` ndx ) , .+ >. , <. ( .r ` ndx ) , .x. >. }
5 4 1 sseqtrri
 |-  { <. ( Base ` ndx ) , B >. } C_ R
6 2 3 5 strfv
 |-  ( B e. V -> B = ( Base ` R ) )