Theorem vtxdushgrfvedglem

Description: Lemma for vtxdushgrfvedg and vtxdusgrfvedg . (Contributed by AV, 12-Dec-2020) (Proof shortened by AV, 5-May-2021)

	Ref	Expression
Hypotheses	vtxdushgrfvedg.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
	vtxdushgrfvedg.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
Assertion	vtxdushgrfvedglem	⊢ ( ( 𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉 ) → ( ♯ ‘ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑈 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) } ) = ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒 } ) )

Proof

Step	Hyp	Ref	Expression
1		vtxdushgrfvedg.v	⊢ 𝑉 = ( Vtx ‘ 𝐺 )
2		vtxdushgrfvedg.e	⊢ 𝐸 = ( Edg ‘ 𝐺 )
3		fvex	⊢ ( iEdg ‘ 𝐺 ) ∈ V
4	3	dmex	⊢ dom ( iEdg ‘ 𝐺 ) ∈ V
5	4	rabex	⊢ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑈 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) } ∈ V
6	5	a1i	⊢ ( ( 𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉 ) → { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑈 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) } ∈ V )
7		eqid	⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
8		eqid	⊢ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑈 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) } = { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑈 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) }
9		eleq2w	⊢ ( 𝑒 = 𝑐 → ( 𝑈 ∈ 𝑒 ↔ 𝑈 ∈ 𝑐 ) )
10	9	cbvrabv	⊢ { 𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒 } = { 𝑐 ∈ 𝐸 ∣ 𝑈 ∈ 𝑐 }
11		eqid	⊢ ( 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑈 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) } ↦ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) = ( 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑈 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) } ↦ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) )
12	2 7 1 8 10 11	ushgredgedg	⊢ ( ( 𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉 ) → ( 𝑥 ∈ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑈 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) } ↦ ( ( iEdg ‘ 𝐺 ) ‘ 𝑥 ) ) : { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑈 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) } –1-1-onto→ { 𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒 } )
13	6 12	hasheqf1od	⊢ ( ( 𝐺 ∈ USHGraph ∧ 𝑈 ∈ 𝑉 ) → ( ♯ ‘ { 𝑖 ∈ dom ( iEdg ‘ 𝐺 ) ∣ 𝑈 ∈ ( ( iEdg ‘ 𝐺 ) ‘ 𝑖 ) } ) = ( ♯ ‘ { 𝑒 ∈ 𝐸 ∣ 𝑈 ∈ 𝑒 } ) )