Theorem usgredgedg

Description: In a simple graph there is a 1-1 onto mapping between the indexed edges containing a fixed vertex and the set of edges containing this vertex. (Contributed by AV, 18-Oct-2020) (Proof shortened by AV, 11-Dec-2020)

	Ref	Expression
Hypotheses	ushgredgedg.e	$$⊢E=\mathrm{Edg}\left(G\right)$$
	ushgredgedg.i	$$⊢I=\mathrm{iEdg}\left(G\right)$$
	ushgredgedg.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
	ushgredgedg.a	$$⊢A=\left\{i\in \mathrm{dom}I|N\in I\left(i\right)\right\}$$
	ushgredgedg.b	$$⊢B=\left\{e\in E|N\in e\right\}$$
	ushgredgedg.f	$$⊢F=\left(x\in A⟼I\left(x\right)\right)$$
Assertion	usgredgedg	$$⊢\left(G\in \mathrm{USGraph}\wedge N\in V\right)\to F:A\underset{1-1\; onto}{⟶}B$$

Proof

Step	Hyp	Ref	Expression
1		ushgredgedg.e	$$⊢E=\mathrm{Edg}\left(G\right)$$
2		ushgredgedg.i	$$⊢I=\mathrm{iEdg}\left(G\right)$$
3		ushgredgedg.v	$$⊢V=\mathrm{Vtx}\left(G\right)$$
4		ushgredgedg.a	$$⊢A=\left\{i\in \mathrm{dom}I|N\in I\left(i\right)\right\}$$
5		ushgredgedg.b	$$⊢B=\left\{e\in E|N\in e\right\}$$
6		ushgredgedg.f	$$⊢F=\left(x\in A⟼I\left(x\right)\right)$$
7		usgruspgr	$$⊢G\in \mathrm{USGraph}\to G\in \mathrm{USHGraph}$$
8		uspgrushgr	$$⊢G\in \mathrm{USHGraph}\to G\in \mathrm{USHGraph}$$
9	7 8	syl	$$⊢G\in \mathrm{USGraph}\to G\in \mathrm{USHGraph}$$
10	1 2 3 4 5 6	ushgredgedg	$$⊢\left(G\in \mathrm{USHGraph}\wedge N\in V\right)\to F:A\underset{1-1\; onto}{⟶}B$$
11	9 10	sylan	$$⊢\left(G\in \mathrm{USGraph}\wedge N\in V\right)\to F:A\underset{1-1\; onto}{⟶}B$$