Metamath Proof Explorer


Theorem isubgredgss

Description: The edges of an induced subgraph of a graph are edges of the graph. (Contributed by AV, 24-Sep-2025)

Ref Expression
Hypotheses isubgredg.v V = Vtx G
isubgredg.e E = Edg G
isubgredg.h H = G ISubGr S
isubgredg.i I = Edg H
Assertion isubgredgss G W S V I E

Proof

Step Hyp Ref Expression
1 isubgredg.v V = Vtx G
2 isubgredg.e E = Edg G
3 isubgredg.h H = G ISubGr S
4 isubgredg.i I = Edg H
5 3 fveq2i iEdg H = iEdg G ISubGr S
6 eqid iEdg G = iEdg G
7 1 6 isubgriedg G W S V iEdg G ISubGr S = iEdg G x dom iEdg G | iEdg G x S
8 5 7 eqtrid G W S V iEdg H = iEdg G x dom iEdg G | iEdg G x S
9 8 rneqd G W S V ran iEdg H = ran iEdg G x dom iEdg G | iEdg G x S
10 resss iEdg G x dom iEdg G | iEdg G x S iEdg G
11 rnss iEdg G x dom iEdg G | iEdg G x S iEdg G ran iEdg G x dom iEdg G | iEdg G x S ran iEdg G
12 10 11 mp1i G W S V ran iEdg G x dom iEdg G | iEdg G x S ran iEdg G
13 9 12 eqsstrd G W S V ran iEdg H ran iEdg G
14 edgval Edg H = ran iEdg H
15 4 14 eqtri I = ran iEdg H
16 edgval Edg G = ran iEdg G
17 2 16 eqtri E = ran iEdg G
18 13 15 17 3sstr4g G W S V I E