Theorem frgrncvvdeqlem4

Description: Lemma 4 for frgrncvvdeq . The mapping of neighbors to neighbors is a function. (Contributed by Alexander van der Vekens, 22-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	frgrncvvdeqlem4	⊢ ( 𝜑 → 𝐴 : 𝐷 ⟶ 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	1 2 3 4 5 6 7 8 9 10	frgrncvvdeqlem2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ∃! 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 )
12		riotacl	⊢ ( ∃! 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 → ( ℩ 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 ) ∈ 𝑁 )
13	11 12	syl	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ( ℩ 𝑦 ∈ 𝑁 { 𝑥 , 𝑦 } ∈ 𝐸 ) ∈ 𝑁 )
14	13 10	fmptd	⊢ ( 𝜑 → 𝐴 : 𝐷 ⟶ 𝑁 )