Theorem usgrlimprop

Description: Properties of a local isomorphism of simple pseudographs. (Contributed by AV, 17-Aug-2025)

	Ref	Expression
Hypotheses	usgrlimprop.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	usgrlimprop.w	⊢ 𝑊 = ( Vtx ‘ 𝐻 )
	usgrlimprop.n	⊢ 𝑁 = ( 𝐺 ClNeighbVtx 𝑣 )
	usgrlimprop.m	⊢ 𝑀 = ( 𝐻 ClNeighbVtx ( 𝐹 ‘ 𝑣 ) )
	usgrlimprop.i	⊢ 𝐼 = ( Edg ‘ 𝐺 )
	usgrlimprop.j	⊢ 𝐽 = ( Edg ‘ 𝐻 )
	usgrlimprop.k	⊢ 𝐾 = { 𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁 }
	usgrlimprop.l	⊢ 𝐿 = { 𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀 }
Assertion	usgrlimprop	⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 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 ∧ ∀ 𝑒 ∈ 𝐾 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		usgrlimprop.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		usgrlimprop.w	⊢ 𝑊 = ( Vtx ‘ 𝐻 )
3		usgrlimprop.n	⊢ 𝑁 = ( 𝐺 ClNeighbVtx 𝑣 )
4		usgrlimprop.m	⊢ 𝑀 = ( 𝐻 ClNeighbVtx ( 𝐹 ‘ 𝑣 ) )
5		usgrlimprop.i	⊢ 𝐼 = ( Edg ‘ 𝐺 )
6		usgrlimprop.j	⊢ 𝐽 = ( Edg ‘ 𝐻 )
7		usgrlimprop.k	⊢ 𝐾 = { 𝑥 ∈ 𝐼 ∣ 𝑥 ⊆ 𝑁 }
8		usgrlimprop.l	⊢ 𝐿 = { 𝑥 ∈ 𝐽 ∣ 𝑥 ⊆ 𝑀 }
9		simp3	⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) ) → 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) )
10	1 2 3 4 5 6 7 8	uspgrlim	⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 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 ∧ ∀ 𝑒 ∈ 𝐾 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ) ) )
11	9 10	mpbid	⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 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 ∧ ∀ 𝑒 ∈ 𝐾 ( 𝑓 “ 𝑒 ) = ( 𝑔 ‘ 𝑒 ) ) ) ) )