Metamath Proof Explorer


Theorem clwlknf1oclwwlknlem3

Description: Lemma 3 for clwlknf1oclwwlkn : The bijective function of clwlknf1oclwwlkn is the bijective function of clwlkclwwlkf1o restricted to the closed walks with a fixed positive length. (Contributed by AV, 26-May-2022) (Revised by AV, 1-Nov-2022)

Ref Expression
Hypotheses clwlknf1oclwwlkn.a 𝐴 = ( 1st𝑐 )
clwlknf1oclwwlkn.b 𝐵 = ( 2nd𝑐 )
clwlknf1oclwwlkn.c 𝐶 = { 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ∣ ( ♯ ‘ ( 1st𝑤 ) ) = 𝑁 }
clwlknf1oclwwlkn.f 𝐹 = ( 𝑐𝐶 ↦ ( 𝐵 prefix ( ♯ ‘ 𝐴 ) ) )
Assertion clwlknf1oclwwlknlem3 ( ( 𝐺 ∈ USPGraph ∧ 𝑁 ∈ ℕ ) → 𝐹 = ( ( 𝑐 ∈ { 𝑤 ∈ ( ClWalks ‘ 𝐺 ) ∣ 1 ≤ ( ♯ ‘ ( 1st𝑤 ) ) } ↦ ( 𝐵 prefix ( ♯ ‘ 𝐴 ) ) ) ↾ 𝐶 ) )

Proof

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