Metamath Proof Explorer


Theorem 0pth

Description: A pair of an empty set (of edges) and a second set (of vertices) is a path iff the second set contains exactly one vertex. (Contributed by Alexander van der Vekens, 30-Oct-2017) (Revised by AV, 19-Jan-2021) (Revised by AV, 30-Oct-2021)

Ref Expression
Hypothesis 0pth.v 𝑉 = ( Vtx ‘ 𝐺 )
Assertion 0pth ( 𝐺𝑊 → ( ∅ ( Paths ‘ 𝐺 ) 𝑃𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) )

Proof

Step Hyp Ref Expression
1 0pth.v 𝑉 = ( Vtx ‘ 𝐺 )
2 ispth ( ∅ ( Paths ‘ 𝐺 ) 𝑃 ↔ ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ ∅ ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ) = ∅ ) )
3 2 a1i ( 𝐺𝑊 → ( ∅ ( Paths ‘ 𝐺 ) 𝑃 ↔ ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ ∅ ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ) = ∅ ) ) )
4 3anass ( ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ ∅ ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ) = ∅ ) ↔ ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ∧ ( Fun ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ ∅ ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ) = ∅ ) ) )
5 4 a1i ( 𝐺𝑊 → ( ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ ∅ ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ) = ∅ ) ↔ ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ∧ ( Fun ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ ∅ ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ) = ∅ ) ) ) )
6 funcnv0 Fun
7 hash0 ( ♯ ‘ ∅ ) = 0
8 0le1 0 ≤ 1
9 7 8 eqbrtri ( ♯ ‘ ∅ ) ≤ 1
10 1z 1 ∈ ℤ
11 0z 0 ∈ ℤ
12 7 11 eqeltri ( ♯ ‘ ∅ ) ∈ ℤ
13 fzon ( ( 1 ∈ ℤ ∧ ( ♯ ‘ ∅ ) ∈ ℤ ) → ( ( ♯ ‘ ∅ ) ≤ 1 ↔ ( 1 ..^ ( ♯ ‘ ∅ ) ) = ∅ ) )
14 10 12 13 mp2an ( ( ♯ ‘ ∅ ) ≤ 1 ↔ ( 1 ..^ ( ♯ ‘ ∅ ) ) = ∅ )
15 9 14 mpbi ( 1 ..^ ( ♯ ‘ ∅ ) ) = ∅
16 15 reseq2i ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) = ( 𝑃 ↾ ∅ )
17 res0 ( 𝑃 ↾ ∅ ) = ∅
18 16 17 eqtri ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) = ∅
19 18 cnveqi ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) =
20 19 funeqi ( Fun ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ↔ Fun ∅ )
21 6 20 mpbir Fun ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ ∅ ) ) )
22 15 imaeq2i ( 𝑃 “ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) = ( 𝑃 “ ∅ )
23 ima0 ( 𝑃 “ ∅ ) = ∅
24 22 23 eqtri ( 𝑃 “ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) = ∅
25 24 ineq2i ( ( 𝑃 “ { 0 , ( ♯ ‘ ∅ ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ) = ( ( 𝑃 “ { 0 , ( ♯ ‘ ∅ ) } ) ∩ ∅ )
26 in0 ( ( 𝑃 “ { 0 , ( ♯ ‘ ∅ ) } ) ∩ ∅ ) = ∅
27 25 26 eqtri ( ( 𝑃 “ { 0 , ( ♯ ‘ ∅ ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ) = ∅
28 21 27 pm3.2i ( Fun ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ ∅ ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ) = ∅ )
29 28 biantru ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ↔ ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ∧ ( Fun ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ ∅ ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ) = ∅ ) ) )
30 5 29 bitr4di ( 𝐺𝑊 → ( ( ∅ ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ ∅ ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ ∅ ) ) ) ) = ∅ ) ↔ ∅ ( Trails ‘ 𝐺 ) 𝑃 ) )
31 1 0trl ( 𝐺𝑊 → ( ∅ ( Trails ‘ 𝐺 ) 𝑃𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) )
32 3 30 31 3bitrd ( 𝐺𝑊 → ( ∅ ( Paths ‘ 𝐺 ) 𝑃𝑃 : ( 0 ... 0 ) ⟶ 𝑉 ) )