Step |
Hyp |
Ref |
Expression |
1 |
|
clwlknf1oclwwlkn.a |
⊢ 𝐴 = ( 1st ‘ 𝑐 ) |
2 |
|
clwlknf1oclwwlkn.b |
⊢ 𝐵 = ( 2nd ‘ 𝑐 ) |
3 |
|
clwlknf1oclwwlkn.c |
⊢ 𝐶 = { 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ∣ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) = 𝑁 } |
4 |
|
clwlknf1oclwwlkn.f |
⊢ 𝐹 = ( 𝑐 ∈ 𝐶 ↦ ( 𝐵 prefix ( ♯ ‘ 𝐴 ) ) ) |
5 |
|
nnge1 |
⊢ ( 𝑁 ∈ ℕ → 1 ≤ 𝑁 ) |
6 |
|
breq2 |
⊢ ( ( ♯ ‘ ( 1st ‘ 𝑤 ) ) = 𝑁 → ( 1 ≤ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) ↔ 1 ≤ 𝑁 ) ) |
7 |
5 6
|
syl5ibrcom |
⊢ ( 𝑁 ∈ ℕ → ( ( ♯ ‘ ( 1st ‘ 𝑤 ) ) = 𝑁 → 1 ≤ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) ) ) |
8 |
7
|
ad2antlr |
⊢ ( ( ( 𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ ) ∧ 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ) → ( ( ♯ ‘ ( 1st ‘ 𝑤 ) ) = 𝑁 → 1 ≤ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) ) ) |
9 |
8
|
ss2rabdv |
⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ ) → { 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ∣ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) = 𝑁 } ⊆ { 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ∣ 1 ≤ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) } ) |
10 |
3 9
|
eqsstrid |
⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ ) → 𝐶 ⊆ { 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ∣ 1 ≤ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) } ) |
11 |
10
|
resmptd |
⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ ) → ( ( 𝑐 ∈ { 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ∣ 1 ≤ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) } ↦ ( 𝐵 prefix ( ♯ ‘ 𝐴 ) ) ) ↾ 𝐶 ) = ( 𝑐 ∈ 𝐶 ↦ ( 𝐵 prefix ( ♯ ‘ 𝐴 ) ) ) ) |
12 |
4 11
|
eqtr4id |
⊢ ( ( 𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ ) → 𝐹 = ( ( 𝑐 ∈ { 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ∣ 1 ≤ ( ♯ ‘ ( 1st ‘ 𝑤 ) ) } ↦ ( 𝐵 prefix ( ♯ ‘ 𝐴 ) ) ) ↾ 𝐶 ) ) |