Theorem elwwlks2on

Description: A walk of length 2 between two vertices as length 3 string. (Contributed by Alexander van der Vekens, 15-Feb-2018) (Revised by AV, 12-May-2021)

	Ref	Expression
Hypothesis	elwwlks2on.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
Assertion	elwwlks2on	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( 𝑊 ∈ ( 𝐴 ( 2 WWalksNOn 𝐺 ) 𝐶 ) ↔ ∃ 𝑏 ∈ 𝑉 ( 𝑊 = ⟨“ 𝐴 𝑏 𝐶 ”⟩ ∧ ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) 𝑊 ∧ ( ♯ ‘ 𝑓 ) = 2 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		elwwlks2on.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2	1	elwwlks2ons3	⊢ ( 𝑊 ∈ ( 𝐴 ( 2 WWalksNOn 𝐺 ) 𝐶 ) ↔ ∃ 𝑏 ∈ 𝑉 ( 𝑊 = ⟨“ 𝐴 𝑏 𝐶 ”⟩ ∧ ⟨“ 𝐴 𝑏 𝐶 ”⟩ ∈ ( 𝐴 ( 2 WWalksNOn 𝐺 ) 𝐶 ) ) )
3	1	s3wwlks2on	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( ⟨“ 𝐴 𝑏 𝐶 ”⟩ ∈ ( 𝐴 ( 2 WWalksNOn 𝐺 ) 𝐶 ) ↔ ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝑏 𝐶 ”⟩ ∧ ( ♯ ‘ 𝑓 ) = 2 ) ) )
4		breq2	⊢ ( ⟨“ 𝐴 𝑏 𝐶 ”⟩ = 𝑊 → ( 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝑏 𝐶 ”⟩ ↔ 𝑓 ( Walks ‘ 𝐺 ) 𝑊 ) )
5	4	eqcoms	⊢ ( 𝑊 = ⟨“ 𝐴 𝑏 𝐶 ”⟩ → ( 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝑏 𝐶 ”⟩ ↔ 𝑓 ( Walks ‘ 𝐺 ) 𝑊 ) )
6	5	anbi1d	⊢ ( 𝑊 = ⟨“ 𝐴 𝑏 𝐶 ”⟩ → ( ( 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝑏 𝐶 ”⟩ ∧ ( ♯ ‘ 𝑓 ) = 2 ) ↔ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑊 ∧ ( ♯ ‘ 𝑓 ) = 2 ) ) )
7	6	exbidv	⊢ ( 𝑊 = ⟨“ 𝐴 𝑏 𝐶 ”⟩ → ( ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) ⟨“ 𝐴 𝑏 𝐶 ”⟩ ∧ ( ♯ ‘ 𝑓 ) = 2 ) ↔ ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) 𝑊 ∧ ( ♯ ‘ 𝑓 ) = 2 ) ) )
8	3 7	sylan9bb	⊢ ( ( ( 𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) ∧ 𝑊 = ⟨“ 𝐴 𝑏 𝐶 ”⟩ ) → ( ⟨“ 𝐴 𝑏 𝐶 ”⟩ ∈ ( 𝐴 ( 2 WWalksNOn 𝐺 ) 𝐶 ) ↔ ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) 𝑊 ∧ ( ♯ ‘ 𝑓 ) = 2 ) ) )
9	8	pm5.32da	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( ( 𝑊 = ⟨“ 𝐴 𝑏 𝐶 ”⟩ ∧ ⟨“ 𝐴 𝑏 𝐶 ”⟩ ∈ ( 𝐴 ( 2 WWalksNOn 𝐺 ) 𝐶 ) ) ↔ ( 𝑊 = ⟨“ 𝐴 𝑏 𝐶 ”⟩ ∧ ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) 𝑊 ∧ ( ♯ ‘ 𝑓 ) = 2 ) ) ) )
10	9	rexbidv	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( ∃ 𝑏 ∈ 𝑉 ( 𝑊 = ⟨“ 𝐴 𝑏 𝐶 ”⟩ ∧ ⟨“ 𝐴 𝑏 𝐶 ”⟩ ∈ ( 𝐴 ( 2 WWalksNOn 𝐺 ) 𝐶 ) ) ↔ ∃ 𝑏 ∈ 𝑉 ( 𝑊 = ⟨“ 𝐴 𝑏 𝐶 ”⟩ ∧ ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) 𝑊 ∧ ( ♯ ‘ 𝑓 ) = 2 ) ) ) )
11	2 10	syl5bb	⊢ ( ( 𝐺 ∈ UPGraph ∧ 𝐴 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( 𝑊 ∈ ( 𝐴 ( 2 WWalksNOn 𝐺 ) 𝐶 ) ↔ ∃ 𝑏 ∈ 𝑉 ( 𝑊 = ⟨“ 𝐴 𝑏 𝐶 ”⟩ ∧ ∃ 𝑓 ( 𝑓 ( Walks ‘ 𝐺 ) 𝑊 ∧ ( ♯ ‘ 𝑓 ) = 2 ) ) ) )