Metamath Proof Explorer


Theorem stgr1

Description: The star graph S_1 consists of a single simple edge. (Contributed by AV, 11-Sep-2025)

Ref Expression
Assertion stgr1 ( StarGr ‘ 1 ) = { ⟨ ( Base ‘ ndx ) , { 0 , 1 } ⟩ , ⟨ ( .ef ‘ ndx ) , ( I ↾ { { 0 , 1 } } ) ⟩ }

Proof

Step Hyp Ref Expression
1 1nn0 1 ∈ ℕ0
2 stgrfv ( 1 ∈ ℕ0 → ( StarGr ‘ 1 ) = { ⟨ ( Base ‘ ndx ) , ( 0 ... 1 ) ⟩ , ⟨ ( .ef ‘ ndx ) , ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∣ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } } ) ⟩ } )
3 1 2 ax-mp ( StarGr ‘ 1 ) = { ⟨ ( Base ‘ ndx ) , ( 0 ... 1 ) ⟩ , ⟨ ( .ef ‘ ndx ) , ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∣ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } } ) ⟩ }
4 fz01pr ( 0 ... 1 ) = { 0 , 1 }
5 4 opeq2i ⟨ ( Base ‘ ndx ) , ( 0 ... 1 ) ⟩ = ⟨ ( Base ‘ ndx ) , { 0 , 1 } ⟩
6 elsni ( 𝑥 ∈ { 1 } → 𝑥 = 1 )
7 preq2 ( 𝑥 = 1 → { 0 , 𝑥 } = { 0 , 1 } )
8 7 eqeq2d ( 𝑥 = 1 → ( 𝑒 = { 0 , 𝑥 } ↔ 𝑒 = { 0 , 1 } ) )
9 8 biimpd ( 𝑥 = 1 → ( 𝑒 = { 0 , 𝑥 } → 𝑒 = { 0 , 1 } ) )
10 6 9 syl ( 𝑥 ∈ { 1 } → ( 𝑒 = { 0 , 𝑥 } → 𝑒 = { 0 , 1 } ) )
11 1z 1 ∈ ℤ
12 fzsn ( 1 ∈ ℤ → ( 1 ... 1 ) = { 1 } )
13 11 12 ax-mp ( 1 ... 1 ) = { 1 }
14 10 13 eleq2s ( 𝑥 ∈ ( 1 ... 1 ) → ( 𝑒 = { 0 , 𝑥 } → 𝑒 = { 0 , 1 } ) )
15 14 rexlimiv ( ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } → 𝑒 = { 0 , 1 } )
16 15 adantl ( ( 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∧ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } ) → 𝑒 = { 0 , 1 } )
17 c0ex 0 ∈ V
18 17 prid1 0 ∈ { 0 , 1 }
19 18 4 eleqtrri 0 ∈ ( 0 ... 1 )
20 1ex 1 ∈ V
21 20 prid2 1 ∈ { 0 , 1 }
22 21 4 eleqtrri 1 ∈ ( 0 ... 1 )
23 prelpwi ( ( 0 ∈ ( 0 ... 1 ) ∧ 1 ∈ ( 0 ... 1 ) ) → { 0 , 1 } ∈ 𝒫 ( 0 ... 1 ) )
24 19 22 23 mp2an { 0 , 1 } ∈ 𝒫 ( 0 ... 1 )
25 eqid { 0 , 1 } = { 0 , 1 }
26 13 rexeqi ( ∃ 𝑥 ∈ ( 1 ... 1 ) { 0 , 1 } = { 0 , 𝑥 } ↔ ∃ 𝑥 ∈ { 1 } { 0 , 1 } = { 0 , 𝑥 } )
27 7 eqeq2d ( 𝑥 = 1 → ( { 0 , 1 } = { 0 , 𝑥 } ↔ { 0 , 1 } = { 0 , 1 } ) )
28 20 27 rexsn ( ∃ 𝑥 ∈ { 1 } { 0 , 1 } = { 0 , 𝑥 } ↔ { 0 , 1 } = { 0 , 1 } )
29 26 28 bitri ( ∃ 𝑥 ∈ ( 1 ... 1 ) { 0 , 1 } = { 0 , 𝑥 } ↔ { 0 , 1 } = { 0 , 1 } )
30 25 29 mpbir 𝑥 ∈ ( 1 ... 1 ) { 0 , 1 } = { 0 , 𝑥 }
31 24 30 pm3.2i ( { 0 , 1 } ∈ 𝒫 ( 0 ... 1 ) ∧ ∃ 𝑥 ∈ ( 1 ... 1 ) { 0 , 1 } = { 0 , 𝑥 } )
32 eleq1 ( 𝑒 = { 0 , 1 } → ( 𝑒 ∈ 𝒫 ( 0 ... 1 ) ↔ { 0 , 1 } ∈ 𝒫 ( 0 ... 1 ) ) )
33 eqeq1 ( 𝑒 = { 0 , 1 } → ( 𝑒 = { 0 , 𝑥 } ↔ { 0 , 1 } = { 0 , 𝑥 } ) )
34 33 rexbidv ( 𝑒 = { 0 , 1 } → ( ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } ↔ ∃ 𝑥 ∈ ( 1 ... 1 ) { 0 , 1 } = { 0 , 𝑥 } ) )
35 32 34 anbi12d ( 𝑒 = { 0 , 1 } → ( ( 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∧ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } ) ↔ ( { 0 , 1 } ∈ 𝒫 ( 0 ... 1 ) ∧ ∃ 𝑥 ∈ ( 1 ... 1 ) { 0 , 1 } = { 0 , 𝑥 } ) ) )
36 31 35 mpbiri ( 𝑒 = { 0 , 1 } → ( 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∧ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } ) )
37 16 36 impbii ( ( 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∧ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } ) ↔ 𝑒 = { 0 , 1 } )
38 37 abbii { 𝑒 ∣ ( 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∧ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } ) } = { 𝑒𝑒 = { 0 , 1 } }
39 df-rab { 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∣ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } } = { 𝑒 ∣ ( 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∧ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } ) }
40 df-sn { { 0 , 1 } } = { 𝑒𝑒 = { 0 , 1 } }
41 38 39 40 3eqtr4i { 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∣ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } } = { { 0 , 1 } }
42 41 reseq2i ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∣ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } } ) = ( I ↾ { { 0 , 1 } } )
43 42 opeq2i ⟨ ( .ef ‘ ndx ) , ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∣ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } } ) ⟩ = ⟨ ( .ef ‘ ndx ) , ( I ↾ { { 0 , 1 } } ) ⟩
44 5 43 preq12i { ⟨ ( Base ‘ ndx ) , ( 0 ... 1 ) ⟩ , ⟨ ( .ef ‘ ndx ) , ( I ↾ { 𝑒 ∈ 𝒫 ( 0 ... 1 ) ∣ ∃ 𝑥 ∈ ( 1 ... 1 ) 𝑒 = { 0 , 𝑥 } } ) ⟩ } = { ⟨ ( Base ‘ ndx ) , { 0 , 1 } ⟩ , ⟨ ( .ef ‘ ndx ) , ( I ↾ { { 0 , 1 } } ) ⟩ }
45 3 44 eqtri ( StarGr ‘ 1 ) = { ⟨ ( Base ‘ ndx ) , { 0 , 1 } ⟩ , ⟨ ( .ef ‘ ndx ) , ( I ↾ { { 0 , 1 } } ) ⟩ }