Theorem grlimprop2

Description: Properties of a local isomorphism of graphs. (Contributed by AV, 29-May-2025)

	Ref	Expression
Hypotheses	grlimprop.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	grlimprop.w	⊢ 𝑊 = ( Vtx ‘ 𝐻 )
	grlimprop2.n	⊢ 𝑁 = ( 𝐺 ClNeighbVtx 𝑣 )
	grlimprop2.m	⊢ 𝑀 = ( 𝐻 ClNeighbVtx ( 𝐹 ‘ 𝑣 ) )
	grlimprop2.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
	grlimprop2.j	⊢ 𝐽 = ( iEdg ‘ 𝐻 )
	grlimprop2.k	⊢ 𝐾 = { 𝑥 ∈ dom 𝐼 ∣ ( 𝐼 ‘ 𝑥 ) ⊆ 𝑁 }
	grlimprop2.l	⊢ 𝐿 = { 𝑥 ∈ dom 𝐽 ∣ ( 𝐽 ‘ 𝑥 ) ⊆ 𝑀 }
Assertion	grlimprop2	⊢ ( 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) → ( 𝐹 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑣 ∈ 𝑉 ∃ 𝑓 ( 𝑓 : 𝑁 –1-1-onto→ 𝑀 ∧ ∃ 𝑔 ( 𝑔 : 𝐾 –1-1-onto→ 𝐿 ∧ ∀ 𝑖 ∈ 𝐾 ( 𝑓 “ ( 𝐼 ‘ 𝑖 ) ) = ( 𝐽 ‘ ( 𝑔 ‘ 𝑖 ) ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		grlimprop.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		grlimprop.w	⊢ 𝑊 = ( Vtx ‘ 𝐻 )
3		grlimprop2.n	⊢ 𝑁 = ( 𝐺 ClNeighbVtx 𝑣 )
4		grlimprop2.m	⊢ 𝑀 = ( 𝐻 ClNeighbVtx ( 𝐹 ‘ 𝑣 ) )
5		grlimprop2.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
6		grlimprop2.j	⊢ 𝐽 = ( iEdg ‘ 𝐻 )
7		grlimprop2.k	⊢ 𝐾 = { 𝑥 ∈ dom 𝐼 ∣ ( 𝐼 ‘ 𝑥 ) ⊆ 𝑁 }
8		grlimprop2.l	⊢ 𝐿 = { 𝑥 ∈ dom 𝐽 ∣ ( 𝐽 ‘ 𝑥 ) ⊆ 𝑀 }
9		grlimdmrel	⊢ Rel dom GraphLocIso
10	9	ovrcl	⊢ ( 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) → ( 𝐺 ∈ V ∧ 𝐻 ∈ V ) )
11		id	⊢ ( 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) → 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) )
12		df-3an	⊢ ( ( 𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) ) ↔ ( ( 𝐺 ∈ V ∧ 𝐻 ∈ V ) ∧ 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) ) )
13	10 11 12	sylanbrc	⊢ ( 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) → ( 𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) ) )
14	1 2 3 4 5 6 7 8	isgrlim2	⊢ ( ( 𝐺 ∈ V ∧ 𝐻 ∈ V ∧ 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) ) → ( 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) ↔ ( 𝐹 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑣 ∈ 𝑉 ∃ 𝑓 ( 𝑓 : 𝑁 –1-1-onto→ 𝑀 ∧ ∃ 𝑔 ( 𝑔 : 𝐾 –1-1-onto→ 𝐿 ∧ ∀ 𝑖 ∈ 𝐾 ( 𝑓 “ ( 𝐼 ‘ 𝑖 ) ) = ( 𝐽 ‘ ( 𝑔 ‘ 𝑖 ) ) ) ) ) ) )
15	13 14	syl	⊢ ( 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) → ( 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) ↔ ( 𝐹 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑣 ∈ 𝑉 ∃ 𝑓 ( 𝑓 : 𝑁 –1-1-onto→ 𝑀 ∧ ∃ 𝑔 ( 𝑔 : 𝐾 –1-1-onto→ 𝐿 ∧ ∀ 𝑖 ∈ 𝐾 ( 𝑓 “ ( 𝐼 ‘ 𝑖 ) ) = ( 𝐽 ‘ ( 𝑔 ‘ 𝑖 ) ) ) ) ) ) )
16	15	ibi	⊢ ( 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) → ( 𝐹 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑣 ∈ 𝑉 ∃ 𝑓 ( 𝑓 : 𝑁 –1-1-onto→ 𝑀 ∧ ∃ 𝑔 ( 𝑔 : 𝐾 –1-1-onto→ 𝐿 ∧ ∀ 𝑖 ∈ 𝐾 ( 𝑓 “ ( 𝐼 ‘ 𝑖 ) ) = ( 𝐽 ‘ ( 𝑔 ‘ 𝑖 ) ) ) ) ) )