Theorem frgrncvvdeqlem6

Description: Lemma 6 for frgrncvvdeq . (Contributed by Alexander van der Vekens, 23-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	frgrncvvdeqlem6	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → { 𝑥 , ( 𝐴 ‘ 𝑥 ) } ∈ 𝐸 )

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	frgrncvvdeqlem5	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → { ( 𝐴 ‘ 𝑥 ) } = ( ( 𝐺 NeighbVtx 𝑥 ) ∩ 𝑁 ) )
12		fvex	⊢ ( 𝐴 ‘ 𝑥 ) ∈ V
13		elinsn	⊢ ( ( ( 𝐴 ‘ 𝑥 ) ∈ V ∧ ( ( 𝐺 NeighbVtx 𝑥 ) ∩ 𝑁 ) = { ( 𝐴 ‘ 𝑥 ) } ) → ( ( 𝐴 ‘ 𝑥 ) ∈ ( 𝐺 NeighbVtx 𝑥 ) ∧ ( 𝐴 ‘ 𝑥 ) ∈ 𝑁 ) )
14	12 13	mpan	⊢ ( ( ( 𝐺 NeighbVtx 𝑥 ) ∩ 𝑁 ) = { ( 𝐴 ‘ 𝑥 ) } → ( ( 𝐴 ‘ 𝑥 ) ∈ ( 𝐺 NeighbVtx 𝑥 ) ∧ ( 𝐴 ‘ 𝑥 ) ∈ 𝑁 ) )
15		frgrusgr	⊢ ( 𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph )
16	2	nbusgreledg	⊢ ( 𝐺 ∈ USGraph → ( ( 𝐴 ‘ 𝑥 ) ∈ ( 𝐺 NeighbVtx 𝑥 ) ↔ { ( 𝐴 ‘ 𝑥 ) , 𝑥 } ∈ 𝐸 ) )
17		prcom	⊢ { ( 𝐴 ‘ 𝑥 ) , 𝑥 } = { 𝑥 , ( 𝐴 ‘ 𝑥 ) }
18	17	eleq1i	⊢ ( { ( 𝐴 ‘ 𝑥 ) , 𝑥 } ∈ 𝐸 ↔ { 𝑥 , ( 𝐴 ‘ 𝑥 ) } ∈ 𝐸 )
19	16 18	bitrdi	⊢ ( 𝐺 ∈ USGraph → ( ( 𝐴 ‘ 𝑥 ) ∈ ( 𝐺 NeighbVtx 𝑥 ) ↔ { 𝑥 , ( 𝐴 ‘ 𝑥 ) } ∈ 𝐸 ) )
20	19	biimpd	⊢ ( 𝐺 ∈ USGraph → ( ( 𝐴 ‘ 𝑥 ) ∈ ( 𝐺 NeighbVtx 𝑥 ) → { 𝑥 , ( 𝐴 ‘ 𝑥 ) } ∈ 𝐸 ) )
21	9 15 20	3syl	⊢ ( 𝜑 → ( ( 𝐴 ‘ 𝑥 ) ∈ ( 𝐺 NeighbVtx 𝑥 ) → { 𝑥 , ( 𝐴 ‘ 𝑥 ) } ∈ 𝐸 ) )
22	21	adantr	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → ( ( 𝐴 ‘ 𝑥 ) ∈ ( 𝐺 NeighbVtx 𝑥 ) → { 𝑥 , ( 𝐴 ‘ 𝑥 ) } ∈ 𝐸 ) )
23	22	com12	⊢ ( ( 𝐴 ‘ 𝑥 ) ∈ ( 𝐺 NeighbVtx 𝑥 ) → ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → { 𝑥 , ( 𝐴 ‘ 𝑥 ) } ∈ 𝐸 ) )
24	23	adantr	⊢ ( ( ( 𝐴 ‘ 𝑥 ) ∈ ( 𝐺 NeighbVtx 𝑥 ) ∧ ( 𝐴 ‘ 𝑥 ) ∈ 𝑁 ) → ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → { 𝑥 , ( 𝐴 ‘ 𝑥 ) } ∈ 𝐸 ) )
25	14 24	syl	⊢ ( ( ( 𝐺 NeighbVtx 𝑥 ) ∩ 𝑁 ) = { ( 𝐴 ‘ 𝑥 ) } → ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → { 𝑥 , ( 𝐴 ‘ 𝑥 ) } ∈ 𝐸 ) )
26	25	eqcoms	⊢ ( { ( 𝐴 ‘ 𝑥 ) } = ( ( 𝐺 NeighbVtx 𝑥 ) ∩ 𝑁 ) → ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → { 𝑥 , ( 𝐴 ‘ 𝑥 ) } ∈ 𝐸 ) )
27	11 26	mpcom	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐷 ) → { 𝑥 , ( 𝐴 ‘ 𝑥 ) } ∈ 𝐸 )