Theorem cusgrsizeindslem

Description: Lemma for cusgrsizeinds . (Contributed by Alexander van der Vekens, 11-Jan-2018) (Revised by AV, 9-Nov-2020)

	Ref	Expression
Hypotheses	cusgrsizeindb0.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	cusgrsizeindb0.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	cusgrsizeindslem	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉 ) → ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } ) = ( ( ♯ ‘ 𝑉 ) − 1 ) )

Proof

Step	Hyp	Ref	Expression
1		cusgrsizeindb0.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		cusgrsizeindb0.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3		cusgrcplgr	⊢ ( 𝐺 ∈ ComplUSGraph → 𝐺 ∈ ComplGraph )
4	1	nbcplgr	⊢ ( ( 𝐺 ∈ ComplGraph ∧ 𝑁 ∈ 𝑉 ) → ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑉 ∖ { 𝑁 } ) )
5	3 4	sylan	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉 ) → ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑉 ∖ { 𝑁 } ) )
6	5	3adant2	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉 ) → ( 𝐺 NeighbVtx 𝑁 ) = ( 𝑉 ∖ { 𝑁 } ) )
7	6	fveq2d	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉 ) → ( ♯ ‘ ( 𝐺 NeighbVtx 𝑁 ) ) = ( ♯ ‘ ( 𝑉 ∖ { 𝑁 } ) ) )
8		cusgrusgr	⊢ ( 𝐺 ∈ ComplUSGraph → 𝐺 ∈ USGraph )
9	8	anim1i	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑁 ∈ 𝑉 ) → ( 𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉 ) )
10	9	3adant2	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉 ) → ( 𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉 ) )
11	1 2	nbusgrf1o	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉 ) → ∃ 𝑓 𝑓 : ( 𝐺 NeighbVtx 𝑁 ) –1-1-onto→ { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } )
12	10 11	syl	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉 ) → ∃ 𝑓 𝑓 : ( 𝐺 NeighbVtx 𝑁 ) –1-1-onto→ { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } )
13	1 2	nbusgr	⊢ ( 𝐺 ∈ USGraph → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ 𝑉 ∣ { 𝑁 , 𝑛 } ∈ 𝐸 } )
14	8 13	syl	⊢ ( 𝐺 ∈ ComplUSGraph → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ 𝑉 ∣ { 𝑁 , 𝑛 } ∈ 𝐸 } )
15	14	adantr	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ) → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ 𝑉 ∣ { 𝑁 , 𝑛 } ∈ 𝐸 } )
16		rabfi	⊢ ( 𝑉 ∈ Fin → { 𝑛 ∈ 𝑉 ∣ { 𝑁 , 𝑛 } ∈ 𝐸 } ∈ Fin )
17	16	adantl	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ) → { 𝑛 ∈ 𝑉 ∣ { 𝑁 , 𝑛 } ∈ 𝐸 } ∈ Fin )
18	15 17	eqeltrd	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ) → ( 𝐺 NeighbVtx 𝑁 ) ∈ Fin )
19	18	3adant3	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉 ) → ( 𝐺 NeighbVtx 𝑁 ) ∈ Fin )
20	8	anim1i	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ) → ( 𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin ) )
21	1	isfusgr	⊢ ( 𝐺 ∈ FinUSGraph ↔ ( 𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin ) )
22	20 21	sylibr	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ) → 𝐺 ∈ FinUSGraph )
23		fusgrfis	⊢ ( 𝐺 ∈ FinUSGraph → ( Edg ‘ 𝐺 ) ∈ Fin )
24	2 23	eqeltrid	⊢ ( 𝐺 ∈ FinUSGraph → 𝐸 ∈ Fin )
25		rabfi	⊢ ( 𝐸 ∈ Fin → { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } ∈ Fin )
26	24 25	syl	⊢ ( 𝐺 ∈ FinUSGraph → { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } ∈ Fin )
27	22 26	syl	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ) → { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } ∈ Fin )
28	27	3adant3	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉 ) → { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } ∈ Fin )
29		hasheqf1o	⊢ ( ( ( 𝐺 NeighbVtx 𝑁 ) ∈ Fin ∧ { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } ∈ Fin ) → ( ( ♯ ‘ ( 𝐺 NeighbVtx 𝑁 ) ) = ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } ) ↔ ∃ 𝑓 𝑓 : ( 𝐺 NeighbVtx 𝑁 ) –1-1-onto→ { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } ) )
30	19 28 29	syl2anc	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉 ) → ( ( ♯ ‘ ( 𝐺 NeighbVtx 𝑁 ) ) = ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } ) ↔ ∃ 𝑓 𝑓 : ( 𝐺 NeighbVtx 𝑁 ) –1-1-onto→ { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } ) )
31	12 30	mpbird	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉 ) → ( ♯ ‘ ( 𝐺 NeighbVtx 𝑁 ) ) = ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } ) )
32		hashdifsn	⊢ ( ( 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉 ) → ( ♯ ‘ ( 𝑉 ∖ { 𝑁 } ) ) = ( ( ♯ ‘ 𝑉 ) − 1 ) )
33	32	3adant1	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉 ) → ( ♯ ‘ ( 𝑉 ∖ { 𝑁 } ) ) = ( ( ♯ ‘ 𝑉 ) − 1 ) )
34	7 31 33	3eqtr3d	⊢ ( ( 𝐺 ∈ ComplUSGraph ∧ 𝑉 ∈ Fin ∧ 𝑁 ∈ 𝑉 ) → ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } ) = ( ( ♯ ‘ 𝑉 ) − 1 ) )