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. >. } |
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Assertion | rngbase | |- ( B e. V -> B = ( Base ` R ) ) |
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 ) ) |