Metamath Proof Explorer


Theorem lppthon

Description: A loop (which is an edge at index J ) induces a path of length 1 from a vertex to itself in a hypergraph. (Contributed by AV, 1-Feb-2021)

Ref Expression
Hypothesis lppthon.i I=iEdgG
Assertion lppthon GUHGraphJdomIIJ=A⟨“J”⟩APathsOnGA⟨“AA”⟩

Proof

Step Hyp Ref Expression
1 lppthon.i I=iEdgG
2 eqid ⟨“AA”⟩=⟨“AA”⟩
3 eqid ⟨“J”⟩=⟨“J”⟩
4 1 lpvtx GUHGraphJdomIIJ=AAVtxG
5 simpl3 GUHGraphJdomIIJ=AA=AIJ=A
6 eqid A=A
7 eqneqall A=AAAAAIJ
8 6 7 ax-mp AAAAIJ
9 8 adantl GUHGraphJdomIIJ=AAAAAIJ
10 eqid VtxG=VtxG
11 2 3 4 4 5 9 10 1 1pthond GUHGraphJdomIIJ=A⟨“J”⟩APathsOnGA⟨“AA”⟩