Theorem uspgrf1oedg

Description: The edge function of a simple pseudograph is a bijective function onto the edges of the graph. (Contributed by AV, 2-Jan-2020) (Revised by AV, 15-Oct-2020)

	Ref	Expression
Hypothesis	usgrf1o.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
Assertion	uspgrf1oedg	⊢ ( 𝐺 ∈ USPGraph → 𝐸 : dom 𝐸 –1-1-onto→ ( Edg ‘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		usgrf1o.e	⊢ 𝐸 = ( iEdg ‘ 𝐺 )
2		eqid	⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
3	2 1	uspgrf	⊢ ( 𝐺 ∈ USPGraph → 𝐸 : dom 𝐸 –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } )
4		f1f1orn	⊢ ( 𝐸 : dom 𝐸 –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } → 𝐸 : dom 𝐸 –1-1-onto→ ran 𝐸 )
5	1	rneqi	⊢ ran 𝐸 = ran ( iEdg ‘ 𝐺 )
6		edgval	⊢ ( Edg ‘ 𝐺 ) = ran ( iEdg ‘ 𝐺 )
7	5 6	eqtr4i	⊢ ran 𝐸 = ( Edg ‘ 𝐺 )
8		f1oeq3	⊢ ( ran 𝐸 = ( Edg ‘ 𝐺 ) → ( 𝐸 : dom 𝐸 –1-1-onto→ ran 𝐸 ↔ 𝐸 : dom 𝐸 –1-1-onto→ ( Edg ‘ 𝐺 ) ) )
9	7 8	ax-mp	⊢ ( 𝐸 : dom 𝐸 –1-1-onto→ ran 𝐸 ↔ 𝐸 : dom 𝐸 –1-1-onto→ ( Edg ‘ 𝐺 ) )
10	4 9	sylib	⊢ ( 𝐸 : dom 𝐸 –1-1→ { 𝑥 ∈ ( 𝒫 ( Vtx ‘ 𝐺 ) ∖ { ∅ } ) ∣ ( ♯ ‘ 𝑥 ) ≤ 2 } → 𝐸 : dom 𝐸 –1-1-onto→ ( Edg ‘ 𝐺 ) )
11	3 10	syl	⊢ ( 𝐺 ∈ USPGraph → 𝐸 : dom 𝐸 –1-1-onto→ ( Edg ‘ 𝐺 ) )