Theorem frgrncvvdeqlem5

Description: Lemma 5 for frgrncvvdeq . The mapping of neighbors to neighbors applied on a vertex is the intersection of the corresponding neighborhoods. (Contributed by Alexander van der Vekens, 23-Dec-2017) (Revised by AV, 10-May-2021)

	Ref	Expression
Hypotheses	frgrncvvdeq.v1	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	frgrncvvdeq.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
	frgrncvvdeq.nx	⊢ 𝐷 = ( 𝐺 NeighbVtx 𝑋 )
	frgrncvvdeq.ny	⊢ 𝑁 = ( 𝐺 NeighbVtx 𝑌 )
	frgrncvvdeq.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	frgrncvvdeq.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
	frgrncvvdeq.ne	⊢ ( 𝜑 → 𝑋 ≠ 𝑌 )
	frgrncvvdeq.xy	⊢ ( 𝜑 → 𝑌 ∉ 𝐷 )
	frgrncvvdeq.f	⊢ ( 𝜑 → 𝐺 ∈ FriendGraph )
	frgrncvvdeq.a	⊢ 𝐴 = ( 𝑥 ∈ 𝐷 ↦ ( ℩ 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 ) )
Assertion	frgrncvvdeqlem5	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → { ( 𝐴 ‘ 𝑥 ) } = ( ( 𝐺 NeighbVtx 𝑥 ) ∩ 𝑁 ) )

Proof

Step	Hyp	Ref	Expression
1		frgrncvvdeq.v1	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		frgrncvvdeq.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3		frgrncvvdeq.nx	⊢ 𝐷 = ( 𝐺 NeighbVtx 𝑋 )
4		frgrncvvdeq.ny	⊢ 𝑁 = ( 𝐺 NeighbVtx 𝑌 )
5		frgrncvvdeq.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
6		frgrncvvdeq.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
7		frgrncvvdeq.ne	⊢ ( 𝜑 → 𝑋 ≠ 𝑌 )
8		frgrncvvdeq.xy	⊢ ( 𝜑 → 𝑌 ∉ 𝐷 )
9		frgrncvvdeq.f	⊢ ( 𝜑 → 𝐺 ∈ FriendGraph )
10		frgrncvvdeq.a	⊢ 𝐴 = ( 𝑥 ∈ 𝐷 ↦ ( ℩ 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 ) )
11		simpr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → 𝑥 ∈ 𝐷 )
12		riotaex	⊢ ( ℩ 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 ) ∈ V
13	10	fvmpt2	⊢ ( ( 𝑥 ∈ 𝐷 ∧ ( ℩ 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 ) ∈ V ) → ( 𝐴 ‘ 𝑥 ) = ( ℩ 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 ) )
14	11 12 13	sylancl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ( 𝐴 ‘ 𝑥 ) = ( ℩ 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 ) )
15	14	sneqd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → { ( 𝐴 ‘ 𝑥 ) } = { ( ℩ 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 ) } )
16	1 2 3 4 5 6 7 8 9 10	frgrncvvdeqlem3	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → { ( ℩ 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 ) } = ( ( 𝐺 NeighbVtx 𝑥 ) ∩ 𝑁 ) )
17	15 16	eqtrd	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → { ( 𝐴 ‘ 𝑥 ) } = ( ( 𝐺 NeighbVtx 𝑥 ) ∩ 𝑁 ) )