Metamath Proof Explorer


Theorem usgrsizedg

Description: In a simple graph, the size of the edge function is the number of the edges of the graph. (Contributed by AV, 4-Jan-2020) (Revised by AV, 7-Jun-2021)

Ref Expression
Assertion usgrsizedg G USGraph iEdg G = Edg G

Proof

Step Hyp Ref Expression
1 fvex iEdg G V
2 1 dmex dom iEdg G V
3 eqid Vtx G = Vtx G
4 eqid iEdg G = iEdg G
5 3 4 usgrf G USGraph iEdg G : dom iEdg G 1-1 x 𝒫 Vtx G | x = 2
6 hashf1rn dom iEdg G V iEdg G : dom iEdg G 1-1 x 𝒫 Vtx G | x = 2 iEdg G = ran iEdg G
7 2 5 6 sylancr G USGraph iEdg G = ran iEdg G
8 edgval Edg G = ran iEdg G
9 8 a1i G USGraph Edg G = ran iEdg G
10 9 fveq2d G USGraph Edg G = ran iEdg G
11 7 10 eqtr4d G USGraph iEdg G = Edg G