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Metamath Proof Explorer

Theorem isgrlim2

Description: A local isomorphism of graphs is a bijection between their vertices that preserves neighborhoods. Definitions expanded. (Contributed by AV, 29-May-2025)

Ref Expression
Hypotheses isgrlim.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
isgrlim.w ⊢ 𝑊 = ( Vtx ‘ 𝐻 )
isgrlim2.n ⊢ 𝑁 = ( 𝐺 ClNeighbVtx 𝑣 )
isgrlim2.m ⊢ 𝑀 = ( 𝐻 ClNeighbVtx ( 𝐹 ‘ 𝑣 ) )
isgrlim2.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
isgrlim2.j ⊢ 𝐽 = ( iEdg ‘ 𝐻 )
isgrlim2.k ⊢ 𝐾 = { 𝑥 ∈ dom 𝐼 ∣ ( 𝐼 ‘ 𝑥 ) ⊆ 𝑁 }
isgrlim2.l ⊢ 𝐿 = { 𝑥 ∈ dom 𝐽 ∣ ( 𝐽 ‘ 𝑥 ) ⊆ 𝑀 }
Assertion isgrlim2 ( ( 𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍 ) → ( 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 isgrlim.v ⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2 isgrlim.w ⊢ 𝑊 = ( Vtx ‘ 𝐻 )
3 isgrlim2.n ⊢ 𝑁 = ( 𝐺 ClNeighbVtx 𝑣 )
4 isgrlim2.m ⊢ 𝑀 = ( 𝐻 ClNeighbVtx ( 𝐹 ‘ 𝑣 ) )
5 isgrlim2.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
6 isgrlim2.j ⊢ 𝐽 = ( iEdg ‘ 𝐻 )
7 isgrlim2.k ⊢ 𝐾 = { 𝑥 ∈ dom 𝐼 ∣ ( 𝐼 ‘ 𝑥 ) ⊆ 𝑁 }
8 isgrlim2.l ⊢ 𝐿 = { 𝑥 ∈ dom 𝐽 ∣ ( 𝐽 ‘ 𝑥 ) ⊆ 𝑀 }
9 1 2 isgrlim ⊢ ( ( 𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍 ) → ( 𝐹 ∈ ( 𝐺 GraphLocIso 𝐻 ) ↔ ( 𝐹 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑣 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝐹 ‘ 𝑣 ) ) ) ) ) )
10 3 eqcomi ⊢ ( 𝐺 ClNeighbVtx 𝑣 ) = 𝑁
11 10 oveq2i ⊢ ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑣 ) ) = ( 𝐺 ISubGr 𝑁 )
12 4 eqcomi ⊢ ( 𝐻 ClNeighbVtx ( 𝐹 ‘ 𝑣 ) ) = 𝑀
13 12 oveq2i ⊢ ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝐹 ‘ 𝑣 ) ) ) = ( 𝐻 ISubGr 𝑀 )
14 11 13 breq12i ⊢ ( ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑣 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝐹 ‘ 𝑣 ) ) ) ↔ ( 𝐺 ISubGr 𝑁 ) ≃𝑔𝑟 ( 𝐻 ISubGr 𝑀 ) )
15 14 a1i ⊢ ( ( 𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍 ) → ( ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑣 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝐹 ‘ 𝑣 ) ) ) ↔ ( 𝐺 ISubGr 𝑁 ) ≃𝑔𝑟 ( 𝐻 ISubGr 𝑀 ) ) )
16 5 6 3 4 7 8 clnbgrisubgrgrim ⊢ ( ( 𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ) → ( ( 𝐺 ISubGr 𝑁 ) ≃𝑔𝑟 ( 𝐻 ISubGr 𝑀 ) ↔ ∃ 𝑓 ( 𝑓 : 𝑁 –1-1-onto→ 𝑀 ∧ ∃ 𝑔 ( 𝑔 : 𝐾 –1-1-onto→ 𝐿 ∧ ∀ 𝑖 ∈ 𝐾 ( 𝑓 “ ( 𝐼 ‘ 𝑖 ) ) = ( 𝐽 ‘ ( 𝑔 ‘ 𝑖 ) ) ) ) ) )
17 16 3adant3 ⊢ ( ( 𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍 ) → ( ( 𝐺 ISubGr 𝑁 ) ≃𝑔𝑟 ( 𝐻 ISubGr 𝑀 ) ↔ ∃ 𝑓 ( 𝑓 : 𝑁 –1-1-onto→ 𝑀 ∧ ∃ 𝑔 ( 𝑔 : 𝐾 –1-1-onto→ 𝐿 ∧ ∀ 𝑖 ∈ 𝐾 ( 𝑓 “ ( 𝐼 ‘ 𝑖 ) ) = ( 𝐽 ‘ ( 𝑔 ‘ 𝑖 ) ) ) ) ) )
18 15 17 bitrd ⊢ ( ( 𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍 ) → ( ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑣 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝐹 ‘ 𝑣 ) ) ) ↔ ∃ 𝑓 ( 𝑓 : 𝑁 –1-1-onto→ 𝑀 ∧ ∃ 𝑔 ( 𝑔 : 𝐾 –1-1-onto→ 𝐿 ∧ ∀ 𝑖 ∈ 𝐾 ( 𝑓 “ ( 𝐼 ‘ 𝑖 ) ) = ( 𝐽 ‘ ( 𝑔 ‘ 𝑖 ) ) ) ) ) )
19 18 ralbidv ⊢ ( ( 𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍 ) → ( ∀ 𝑣 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑣 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝐹 ‘ 𝑣 ) ) ) ↔ ∀ 𝑣 ∈ 𝑉 ∃ 𝑓 ( 𝑓 : 𝑁 –1-1-onto→ 𝑀 ∧ ∃ 𝑔 ( 𝑔 : 𝐾 –1-1-onto→ 𝐿 ∧ ∀ 𝑖 ∈ 𝐾 ( 𝑓 “ ( 𝐼 ‘ 𝑖 ) ) = ( 𝐽 ‘ ( 𝑔 ‘ 𝑖 ) ) ) ) ) )
20 19 anbi2d ⊢ ( ( 𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍 ) → ( ( 𝐹 : 𝑉 –1-1-onto→ 𝑊 ∧ ∀ 𝑣 ∈ 𝑉 ( 𝐺 ISubGr ( 𝐺 ClNeighbVtx 𝑣 ) ) ≃𝑔𝑟 ( 𝐻 ISubGr ( 𝐻 ClNeighbVtx ( 𝐹 ‘ 𝑣 ) ) ) ) ↔ ( 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 ) ) ) ) ) ) )
21 9 20 bitrd ⊢ ( ( 𝐺 ∈ 𝑋 ∧ 𝐻 ∈ 𝑌 ∧ 𝐹 ∈ 𝑍 ) → ( 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 ) ) ) ) ) ) )