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 ) , · ⟩ }
Assertion	rngbase	⊢ ( 𝐵 ∈ 𝑉 → 𝐵 = ( Base ‘ 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		rngfn.r	⊢ 𝑅 = { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ }
2	1	rngstr	⊢ 𝑅 Struct ⟨ 1 , 3 ⟩
3		baseid	⊢ Base = Slot ( Base ‘ ndx )
4		snsstp1	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ } ⊆ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ , ⟨ ( +g ‘ ndx ) , + ⟩ , ⟨ ( .r ‘ ndx ) , · ⟩ }
5	4 1	sseqtrri	⊢ { ⟨ ( Base ‘ ndx ) , 𝐵 ⟩ } ⊆ 𝑅
6	2 3 5	strfv	⊢ ( 𝐵 ∈ 𝑉 → 𝐵 = ( Base ‘ 𝑅 ) )