Theorem grlimf1o

Description: A local isomorphism of graphs is a bijection between their vertices. (Contributed by AV, 21-May-2025)

	Ref	Expression
Hypotheses	grlimprop.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	grlimprop.w	⊢ 𝑊 = ( Vtx ‘ 𝐻 )
Assertion	grlimf1o	⊢ ( 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) → 𝐹 : 𝑉 –1-1-onto→ 𝑊 )

Proof

Step	Hyp	Ref	Expression
1		grlimprop.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		grlimprop.w	⊢ 𝑊 = ( Vtx ‘ 𝐻 )
3	1 2	grlimprop	⊢ ( 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) → ( 𝐹 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑣 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝐹 ‘ 𝑣 ) ) ) ) )
4	3	simpld	⊢ ( 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) → 𝐹 : 𝑉 –1-1-onto→ 𝑊 )