Theorem isewlk

Description: Conditions for a function (sequence of hyperedges) to be an s-walk of edges. (Contributed by AV, 4-Jan-2021)

	Ref	Expression
Hypothesis	ewlksfval.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
Assertion	isewlk	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0* ∧ 𝐹 ∈ 𝑈 ) → ( 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		ewlksfval.i	⊢ 𝐼 = ( iEdg ‘ 𝐺 )
2	1	ewlksfval	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0* ) → ( 𝐺 EdgWalks 𝑆 ) = { 𝑓 ∣ ( 𝑓 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝑓 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) ) } )
3	2	3adant3	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0* ∧ 𝐹 ∈ 𝑈 ) → ( 𝐺 EdgWalks 𝑆 ) = { 𝑓 ∣ ( 𝑓 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝑓 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) ) } )
4	3	eleq2d	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0* ∧ 𝐹 ∈ 𝑈 ) → ( 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) ↔ 𝐹 ∈ { 𝑓 ∣ ( 𝑓 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝑓 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) ) } ) )
5		eleq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ∈ Word dom 𝐼 ↔ 𝐹 ∈ Word dom 𝐼 ) )
6		fveq2	⊢ ( 𝑓 = 𝐹 → ( ♯ ‘ 𝑓 ) = ( ♯ ‘ 𝐹 ) )
7	6	oveq2d	⊢ ( 𝑓 = 𝐹 → ( 1 ..^ ( ♯ ‘ 𝑓 ) ) = ( 1 ..^ ( ♯ ‘ 𝐹 ) ) )
8		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ ( 𝑘 − 1 ) ) = ( 𝐹 ‘ ( 𝑘 − 1 ) ) )
9	8	fveq2d	⊢ ( 𝑓 = 𝐹 → ( 𝐼 ‘ ( 𝑓 ‘ ( 𝑘 − 1 ) ) ) = ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) )
10		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) )
11	10	fveq2d	⊢ ( 𝑓 = 𝐹 → ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) = ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) )
12	9 11	ineq12d	⊢ ( 𝑓 = 𝐹 → ( ( 𝐼 ‘ ( 𝑓 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) ) = ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) )
13	12	fveq2d	⊢ ( 𝑓 = 𝐹 → ( ♯ ‘ ( ( 𝐼 ‘ ( 𝑓 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) = ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) )
14	13	breq2d	⊢ ( 𝑓 = 𝐹 → ( 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝑓 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) ↔ 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) )
15	7 14	raleqbidv	⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝑓 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) ↔ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) )
16	5 15	anbi12d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝑓 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) ) ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) )
17	16	elabg	⊢ ( 𝐹 ∈ 𝑈 → ( 𝐹 ∈ { 𝑓 ∣ ( 𝑓 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝑓 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) ) } ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) )
18	17	3ad2ant3	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0* ∧ 𝐹 ∈ 𝑈 ) → ( 𝐹 ∈ { 𝑓 ∣ ( 𝑓 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝑓 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝑓 ‘ 𝑘 ) ) ) ) ) } ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) )
19	4 18	bitrd	⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑆 ∈ ℕ0* ∧ 𝐹 ∈ 𝑈 ) → ( 𝐹 ∈ ( 𝐺 EdgWalks 𝑆 ) ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ ∀ 𝑘 ∈ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) 𝑆 ≤ ( ♯ ‘ ( ( 𝐼 ‘ ( 𝐹 ‘ ( 𝑘 − 1 ) ) ) ∩ ( 𝐼 ‘ ( 𝐹 ‘ 𝑘 ) ) ) ) ) ) )