Theorem 2cycld

Description: Construction of a 2-cycle from two given edges in a graph. (Contributed by BTernaryTau, 16-Oct-2023)

	Ref	Expression
Hypotheses	2cycld.1	⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 ”⟩
	2cycld.2	⊢ 𝐹 = ⟨“ 𝐽 𝐾 ”⟩
	2cycld.3	⊢ ( 𝜑 → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
	2cycld.4	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) )
	2cycld.5	⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
	2cycld.6	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	2cycld.7	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
	2cycld.8	⊢ ( 𝜑 → 𝐽 ≠ 𝐾 )
	2cycld.9	⊢ ( 𝜑 → 𝐴 = 𝐶 )
Assertion	2cycld	⊢ ( 𝜑 → 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 )

Proof

Step	Hyp	Ref	Expression
1		2cycld.1	⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 ”⟩
2		2cycld.2	⊢ 𝐹 = ⟨“ 𝐽 𝐾 ”⟩
3		2cycld.3	⊢ ( 𝜑 → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
4		2cycld.4	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) )
5		2cycld.5	⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
6		2cycld.6	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
7		2cycld.7	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
8		2cycld.8	⊢ ( 𝜑 → 𝐽 ≠ 𝐾 )
9		2cycld.9	⊢ ( 𝜑 → 𝐴 = 𝐶 )
10	1 2 3 4 5 6 7 8	2pthd	⊢ ( 𝜑 → 𝐹 ( Paths ‘ 𝐺 ) 𝑃 )
11	1	fveq1i	⊢ ( 𝑃 ‘ 0 ) = ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 0 )
12		s3fv0	⊢ ( 𝐴 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 0 ) = 𝐴 )
13	11 12	syl5eq	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑃 ‘ 0 ) = 𝐴 )
14	13	3ad2ant1	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( 𝑃 ‘ 0 ) = 𝐴 )
15	14	adantr	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ∧ 𝐴 = 𝐶 ) → ( 𝑃 ‘ 0 ) = 𝐴 )
16		simpr	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ∧ 𝐴 = 𝐶 ) → 𝐴 = 𝐶 )
17	2	fveq2i	⊢ ( ♯ ‘ 𝐹 ) = ( ♯ ‘ ⟨“ 𝐽 𝐾 ”⟩ )
18		s2len	⊢ ( ♯ ‘ ⟨“ 𝐽 𝐾 ”⟩ ) = 2
19	17 18	eqtri	⊢ ( ♯ ‘ 𝐹 ) = 2
20	1 19	fveq12i	⊢ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 2 )
21		s3fv2	⊢ ( 𝐶 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 2 ) = 𝐶 )
22	20 21	syl5req	⊢ ( 𝐶 ∈ 𝑉 → 𝐶 = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) )
23	22	3ad2ant3	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → 𝐶 = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) )
24	23	adantr	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ∧ 𝐴 = 𝐶 ) → 𝐶 = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) )
25	15 16 24	3eqtrd	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ∧ 𝐴 = 𝐶 ) → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) )
26	3 9 25	syl2anc	⊢ ( 𝜑 → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) )
27		iscycl	⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) )
28	10 26 27	sylanbrc	⊢ ( 𝜑 → 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 )