srngbase

Description: The base set of a constructed star ring. (Contributed by Mario Carneiro, 18-Nov-2013) (Revised by Mario Carneiro, 6-May-2015)

		Ref	Expression
	Hypothesis	srngstr.r	$⊢ R = \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, \underset{˙}{+}⟩, ⟨\cdot_{ndx}, \underset{˙}{\cdot}⟩\} \cup \{⟨_{ndx}, \underset{˙}{}⟩\}$
	Assertion	srngbase	$⊢ B \in X \to B = {Base}_{R}$

Step	Hyp	Ref	Expression
1		srngstr.r	$⊢ R = \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, \underset{˙}{+}⟩, ⟨\cdot_{ndx}, \underset{˙}{\cdot}⟩\} \cup \{⟨_{ndx}, \underset{˙}{}⟩\}$
2	1	srngstr	$⊢ R Struct ⟨1, 4⟩$
3		baseid	$⊢ Base = Slot {Base}_{ndx}$
4		snsstp1	$⊢ \{⟨{Base}_{ndx}, B⟩\} \subseteq \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, \underset{˙}{+}⟩, ⟨\cdot_{ndx}, \underset{˙}{\cdot}⟩\}$
5		ssun1	$⊢ \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, \underset{˙}{+}⟩, ⟨\cdot_{ndx}, \underset{˙}{\cdot}⟩\} \subseteq \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, \underset{˙}{+}⟩, ⟨\cdot_{ndx}, \underset{˙}{\cdot}⟩\} \cup \{⟨_{ndx}, \underset{˙}{}⟩\}$
6	5 1	sseqtrri	$⊢ \{⟨{Base}_{ndx}, B⟩, ⟨+_{ndx}, \underset{˙}{+}⟩, ⟨\cdot_{ndx}, \underset{˙}{\cdot}⟩\} \subseteq R$
7	4 6	sstri	$⊢ \{⟨{Base}_{ndx}, B⟩\} \subseteq R$
8	2 3 7	strfv	$⊢ B \in X \to B = {Base}_{R}$

Metamath Proof Explorer