Metamath Proof Explorer


Theorem stgriedg

Description: The indexed edges of the star graph S_N. (Contributed by AV, 11-Sep-2025)

Ref Expression
Assertion stgriedg ( 𝑁 ∈ ℕ0 → ( iEdg ‘ ( StarGr ‘ 𝑁 ) ) = ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 𝑁 ) ∣ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } } ) )

Proof

Step Hyp Ref Expression
1 stgrfv ( 𝑁 ∈ ℕ0 → ( StarGr ‘ 𝑁 ) = { ⟨ ( Base ‘ ndx ) , ( 0 ... 𝑁 ) ⟩ , ⟨ ( .ef ‘ ndx ) , ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 𝑁 ) ∣ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } } ) ⟩ } )
2 1 fveq2d ( 𝑁 ∈ ℕ0 → ( iEdg ‘ ( StarGr ‘ 𝑁 ) ) = ( iEdg ‘ { ⟨ ( Base ‘ ndx ) , ( 0 ... 𝑁 ) ⟩ , ⟨ ( .ef ‘ ndx ) , ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 𝑁 ) ∣ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } } ) ⟩ } ) )
3 ovex ( 0 ... 𝑁 ) ∈ V
4 eqid { 𝑒 ∈ 𝒫 ( 0 ... 𝑁 ) ∣ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } } = { 𝑒 ∈ 𝒫 ( 0 ... 𝑁 ) ∣ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } }
5 pwexg ( ( 0 ... 𝑁 ) ∈ V → 𝒫 ( 0 ... 𝑁 ) ∈ V )
6 4 5 rabexd ( ( 0 ... 𝑁 ) ∈ V → { 𝑒 ∈ 𝒫 ( 0 ... 𝑁 ) ∣ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } } ∈ V )
7 6 resiexd ( ( 0 ... 𝑁 ) ∈ V → ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 𝑁 ) ∣ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } } ) ∈ V )
8 3 7 ax-mp ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 𝑁 ) ∣ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } } ) ∈ V
9 3 8 pm3.2i ( ( 0 ... 𝑁 ) ∈ V ∧ ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 𝑁 ) ∣ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } } ) ∈ V )
10 eqid { ⟨ ( Base ‘ ndx ) , ( 0 ... 𝑁 ) ⟩ , ⟨ ( .ef ‘ ndx ) , ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 𝑁 ) ∣ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } } ) ⟩ } = { ⟨ ( Base ‘ ndx ) , ( 0 ... 𝑁 ) ⟩ , ⟨ ( .ef ‘ ndx ) , ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 𝑁 ) ∣ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } } ) ⟩ }
11 10 struct2griedg ( ( ( 0 ... 𝑁 ) ∈ V ∧ ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 𝑁 ) ∣ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } } ) ∈ V ) → ( iEdg ‘ { ⟨ ( Base ‘ ndx ) , ( 0 ... 𝑁 ) ⟩ , ⟨ ( .ef ‘ ndx ) , ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 𝑁 ) ∣ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } } ) ⟩ } ) = ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 𝑁 ) ∣ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } } ) )
12 9 11 mp1i ( 𝑁 ∈ ℕ0 → ( iEdg ‘ { ⟨ ( Base ‘ ndx ) , ( 0 ... 𝑁 ) ⟩ , ⟨ ( .ef ‘ ndx ) , ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 𝑁 ) ∣ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } } ) ⟩ } ) = ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 𝑁 ) ∣ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } } ) )
13 2 12 eqtrd ( 𝑁 ∈ ℕ0 → ( iEdg ‘ ( StarGr ‘ 𝑁 ) ) = ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 𝑁 ) ∣ ∃ 𝑥 ∈ ( 1 ... 𝑁 ) 𝑒 = { 0 , 𝑥 } } ) )