Theorem dfgrlic3

Description: Alternate, explicit definition of the "is locally isomorphic to" relation for two graphs. (Contributed by AV, 9-Jun-2025)

	Ref	Expression
Hypotheses	dfgrlic2.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	dfgrlic2.w	⊢ 𝑊 = ( Vtx ‘ 𝐻 )
	dfgrlic3.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
	dfgrlic3.j	⊢ 𝐽 = ( iEdg ‘ 𝐻 )
	dfgrlic3.n	⊢ 𝑁 = ( 𝐺 ClNeighbVtx 𝑣 )
	dfgrlic3.m	⊢ 𝑀 = ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑣 ) )
	dfgrlic3.k	⊢ 𝐾 = { 𝑥 ∈ dom 𝐼 ∣ ( 𝐼 ‘ 𝑥 ) ⊆ 𝑁 }
	dfgrlic3.l	⊢ 𝐿 = { 𝑥 ∈ dom 𝐽 ∣ ( 𝐽 ‘ 𝑥 ) ⊆ 𝑀 }
Assertion	dfgrlic3	⊢ ( ( 𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ) → ( 𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃ 𝑓 ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑣 ∈ 𝑉 ∃ 𝑗 ( 𝑗 : 𝑁 –1-1-onto→ 𝑀 ∧ ∃ 𝑔 ( 𝑔 : 𝐾 –1-1-onto→ 𝐿 ∧ ∀ 𝑖 ∈ 𝐾 ( 𝑗 “ ( 𝐼 ‘ 𝑖 ) ) = ( 𝐽 ‘ ( 𝑔 ‘ 𝑖 ) ) ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		dfgrlic2.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		dfgrlic2.w	⊢ 𝑊 = ( Vtx ‘ 𝐻 )
3		dfgrlic3.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
4		dfgrlic3.j	⊢ 𝐽 = ( iEdg ‘ 𝐻 )
5		dfgrlic3.n	⊢ 𝑁 = ( 𝐺 ClNeighbVtx 𝑣 )
6		dfgrlic3.m	⊢ 𝑀 = ( 𝐻 ClNeighbVtx ( 𝑓 ‘ 𝑣 ) )
7		dfgrlic3.k	⊢ 𝐾 = { 𝑥 ∈ dom 𝐼 ∣ ( 𝐼 ‘ 𝑥 ) ⊆ 𝑁 }
8		dfgrlic3.l	⊢ 𝐿 = { 𝑥 ∈ dom 𝐽 ∣ ( 𝐽 ‘ 𝑥 ) ⊆ 𝑀 }
9		brgrlic	⊢ ( 𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ( 𝐺 GraphLocIso 𝐻 ) ≠ ∅ )
10		n0	⊢ ( ( 𝐺 GraphLocIso 𝐻 ) ≠ ∅ ↔ ∃ 𝑓 𝑓 ∈ ( 𝐺 GraphLocIso 𝐻 ) )
11	9 10	bitri	⊢ ( 𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃ 𝑓 𝑓 ∈ ( 𝐺 GraphLocIso 𝐻 ) )
12	1 2 5 6 3 4 7 8	isgrlim2	⊢ ( ( 𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝑓 ∈ V ) → ( 𝑓 ∈ ( 𝐺 GraphLocIso 𝐻 ) ↔ ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑣 ∈ 𝑉 ∃ 𝑗 ( 𝑗 : 𝑁 –1-1-onto→ 𝑀 ∧ ∃ 𝑔 ( 𝑔 : 𝐾 –1-1-onto→ 𝐿 ∧ ∀ 𝑖 ∈ 𝐾 ( 𝑗 “ ( 𝐼 ‘ 𝑖 ) ) = ( 𝐽 ‘ ( 𝑔 ‘ 𝑖 ) ) ) ) ) ) )
13	12	el3v3	⊢ ( ( 𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ) → ( 𝑓 ∈ ( 𝐺 GraphLocIso 𝐻 ) ↔ ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑣 ∈ 𝑉 ∃ 𝑗 ( 𝑗 : 𝑁 –1-1-onto→ 𝑀 ∧ ∃ 𝑔 ( 𝑔 : 𝐾 –1-1-onto→ 𝐿 ∧ ∀ 𝑖 ∈ 𝐾 ( 𝑗 “ ( 𝐼 ‘ 𝑖 ) ) = ( 𝐽 ‘ ( 𝑔 ‘ 𝑖 ) ) ) ) ) ) )
14	13	exbidv	⊢ ( ( 𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ) → ( ∃ 𝑓 𝑓 ∈ ( 𝐺 GraphLocIso 𝐻 ) ↔ ∃ 𝑓 ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑣 ∈ 𝑉 ∃ 𝑗 ( 𝑗 : 𝑁 –1-1-onto→ 𝑀 ∧ ∃ 𝑔 ( 𝑔 : 𝐾 –1-1-onto→ 𝐿 ∧ ∀ 𝑖 ∈ 𝐾 ( 𝑗 “ ( 𝐼 ‘ 𝑖 ) ) = ( 𝐽 ‘ ( 𝑔 ‘ 𝑖 ) ) ) ) ) ) )
15	11 14	bitrid	⊢ ( ( 𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ) → ( 𝐺 ≃𝑙𝑔𝑟 𝐻 ↔ ∃ 𝑓 ( 𝑓 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑣 ∈ 𝑉 ∃ 𝑗 ( 𝑗 : 𝑁 –1-1-onto→ 𝑀 ∧ ∃ 𝑔 ( 𝑔 : 𝐾 –1-1-onto→ 𝐿 ∧ ∀ 𝑖 ∈ 𝐾 ( 𝑗 “ ( 𝐼 ‘ 𝑖 ) ) = ( 𝐽 ‘ ( 𝑔 ‘ 𝑖 ) ) ) ) ) ) )