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 ) )