This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) )