Theorem usgredgleordALT

Description: Alternate proof for usgredgleord based on usgriedgleord . In a simple graph the number of edges which contain a given vertex is not greater than the number of vertices. (Contributed by Alexander van der Vekens, 4-Jan-2018) (Revised by AV, 18-Oct-2020) (Proof shortened by AV, 5-May-2021) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	usgredgleord.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	usgredgleord.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	usgredgleordALT	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉 ) → ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } ) ≤ ( ♯ ‘ 𝑉 ) )

Proof

Step	Hyp	Ref	Expression
1		usgredgleord.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		usgredgleord.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3		fvex	⊢ ( iEdg ‘ 𝐺 ) ∈ V
4	3	dmex	⊢ dom ( iEdg ‘ 𝐺 ) ∈ V
5	4	rabex	⊢ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑁 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) } ∈ V
6	5	a1i	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉 ) → { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑁 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) } ∈ V )
7		eqid	⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
8		eqid	⊢ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑁 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) } = { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑁 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) }
9		eleq2w	⊢ ( 𝑒 = 𝑓 → ( 𝑁 ∈ 𝑒 ↔ 𝑁 ∈ 𝑓 ) )
10	9	cbvrabv	⊢ { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } = { 𝑓 ∈ 𝐸 ∣ 𝑁 ∈ 𝑓 }
11		eqid	⊢ ( 𝑦 ∈ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑁 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) } ↦ ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ) = ( 𝑦 ∈ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑁 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) } ↦ ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) )
12	2 7 1 8 10 11	usgredgedg	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉 ) → ( 𝑦 ∈ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑁 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) } ↦ ( ( iEdg ‘ 𝐺 ) ‘ 𝑦 ) ) : { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑁 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) } –1-1-onto→ { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } )
13	6 12	hasheqf1od	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉 ) → ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑁 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) } ) = ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } ) )
14	1 7	usgriedgleord	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉 ) → ( ♯ ‘ { 𝑥 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑁 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) } ) ≤ ( ♯ ‘ 𝑉 ) )
15	13 14	eqbrtrrd	⊢ ( ( 𝐺 ∈ USGraph ∧ 𝑁 ∈ 𝑉 ) → ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑁 ∈ 𝑒 } ) ≤ ( ♯ ‘ 𝑉 ) )