Metamath Proof Explorer


Theorem struct2grstr

Description: A graph represented as an extensible structure with vertices as base set and indexed edges is actually an extensible structure. (Contributed by AV, 23-Nov-2020)

Ref Expression
Hypothesis struct2grvtx.g 𝐺 = { ⟨ ( Base ‘ ndx ) , 𝑉 ⟩ , ⟨ ( .ef ‘ ndx ) , 𝐸 ⟩ }
Assertion struct2grstr 𝐺 Struct ⟨ ( Base ‘ ndx ) , ( .ef ‘ ndx ) ⟩

Proof

Step Hyp Ref Expression
1 struct2grvtx.g 𝐺 = { ⟨ ( Base ‘ ndx ) , 𝑉 ⟩ , ⟨ ( .ef ‘ ndx ) , 𝐸 ⟩ }
2 baseltedgf ( Base ‘ ndx ) < ( .ef ‘ ndx )
3 edgfndxnn ( .ef ‘ ndx ) ∈ ℕ
4 1 2 3 2strstr1 𝐺 Struct ⟨ ( Base ‘ ndx ) , ( .ef ‘ ndx ) ⟩