Metamath Proof Explorer


Theorem usgrexmpledg

Description: The edges { 0 , 1 } , { 1 , 2 } , { 2 , 0 } , { 0 , 3 } of the graph G = <. V , E >. . (Contributed by AV, 12-Jan-2020) (Revised by AV, 21-Oct-2020)

Ref Expression
Hypotheses usgrexmpl.v V = 0 4
usgrexmpl.e E = ⟨“ 0 1 1 2 2 0 0 3 ”⟩
usgrexmpl.g G = V E
Assertion usgrexmpledg Edg G = 0 1 1 2 2 0 0 3

Proof

Step Hyp Ref Expression
1 usgrexmpl.v V = 0 4
2 usgrexmpl.e E = ⟨“ 0 1 1 2 2 0 0 3 ”⟩
3 usgrexmpl.g G = V E
4 edgval Edg G = ran iEdg G
5 1 2 3 usgrexmpllem Vtx G = V iEdg G = E
6 5 simpri iEdg G = E
7 6 rneqi ran iEdg G = ran E
8 prex 0 1 V
9 prex 1 2 V
10 8 9 pm3.2i 0 1 V 1 2 V
11 prex 2 0 V
12 prex 0 3 V
13 11 12 pm3.2i 2 0 V 0 3 V
14 10 13 pm3.2i 0 1 V 1 2 V 2 0 V 0 3 V
15 usgrexmpldifpr 0 1 1 2 0 1 2 0 0 1 0 3 1 2 2 0 1 2 0 3 2 0 0 3
16 14 15 pm3.2i 0 1 V 1 2 V 2 0 V 0 3 V 0 1 1 2 0 1 2 0 0 1 0 3 1 2 2 0 1 2 0 3 2 0 0 3
17 16 2 pm3.2i 0 1 V 1 2 V 2 0 V 0 3 V 0 1 1 2 0 1 2 0 0 1 0 3 1 2 2 0 1 2 0 3 2 0 0 3 E = ⟨“ 0 1 1 2 2 0 0 3 ”⟩
18 s4f1o 0 1 V 1 2 V 2 0 V 0 3 V 0 1 1 2 0 1 2 0 0 1 0 3 1 2 2 0 1 2 0 3 2 0 0 3 E = ⟨“ 0 1 1 2 2 0 0 3 ”⟩ E : dom E 1-1 onto 0 1 1 2 2 0 0 3
19 18 imp31 0 1 V 1 2 V 2 0 V 0 3 V 0 1 1 2 0 1 2 0 0 1 0 3 1 2 2 0 1 2 0 3 2 0 0 3 E = ⟨“ 0 1 1 2 2 0 0 3 ”⟩ E : dom E 1-1 onto 0 1 1 2 2 0 0 3
20 dff1o5 E : dom E 1-1 onto 0 1 1 2 2 0 0 3 E : dom E 1-1 0 1 1 2 2 0 0 3 ran E = 0 1 1 2 2 0 0 3
21 20 simprbi E : dom E 1-1 onto 0 1 1 2 2 0 0 3 ran E = 0 1 1 2 2 0 0 3
22 17 19 21 mp2b ran E = 0 1 1 2 2 0 0 3
23 4 7 22 3eqtri Edg G = 0 1 1 2 2 0 0 3