Theorem frgrncvvdeqlem10

Description: Lemma 10 for frgrncvvdeq . (Contributed by Alexander van der Vekens, 24-Dec-2017) (Revised by AV, 10-May-2021) (Proof shortened by AV, 30-Dec-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	frgrncvvdeqlem10	⊢ ( 𝜑 → 𝐴 : 𝐷 –1-1-onto→ 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	frgrncvvdeqlem8	⊢ ( 𝜑 → 𝐴 : 𝐷 –1-1→ 𝑁 )
12	1 2 3 4 5 6 7 8 9 10	frgrncvvdeqlem9	⊢ ( 𝜑 → 𝐴 : 𝐷 –onto→ 𝑁 )
13		df-f1o	⊢ ( 𝐴 : 𝐷 –1-1-onto→ 𝑁 ↔ ( 𝐴 : 𝐷 –1-1→ 𝑁 ∧ 𝐴 : 𝐷 –onto→ 𝑁 ) )
14	11 12 13	sylanbrc	⊢ ( 𝜑 → 𝐴 : 𝐷 –1-1-onto→ 𝑁 )