Theorem 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 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ∪ { ⟨ ( *𝑟 ‘ ndx ) , ∗ ⟩ } )
Assertion	srngbase	⊢ ( 𝐵 ∈ 𝑋 → 𝐵 = ( Base ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		srngstr.r	⊢ 𝑅 = ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ∪ { ⟨ ( *𝑟 ‘ ndx ) , ∗ ⟩ } )
2	1	srngstr	⊢ 𝑅 Struct ⟨ 1 , 4 ⟩
3		baseid	⊢ Base = Slot ( Base ‘ ndx )
4		snsstp1	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ }
5		ssun1	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ⊆ ( { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ∪ { ⟨ ( *𝑟 ‘ ndx ) , ∗ ⟩ } )
6	5 1	sseqtrri	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ } ⊆ 𝑅
7	4 6	sstri	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ } ⊆ 𝑅
8	2 3 7	strfv	⊢ ( 𝐵 ∈ 𝑋 → 𝐵 = ( Base ‘ 𝑅 ) )