MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subgrprop2 Structured version   Visualization version   GIF version

Theorem subgrprop2 29291
Description: The properties of a subgraph: If 𝑆 is a subgraph of 𝐺, its vertices are also vertices of 𝐺, and its edges are also edges of 𝐺, connecting vertices of the subgraph only. (Contributed by AV, 19-Nov-2020.)
Hypotheses
Ref Expression
issubgr.v 𝑉 = (Vtx‘𝑆)
issubgr.a 𝐴 = (Vtx‘𝐺)
issubgr.i 𝐼 = (iEdg‘𝑆)
issubgr.b 𝐵 = (iEdg‘𝐺)
issubgr.e 𝐸 = (Edg‘𝑆)
Assertion
Ref Expression
subgrprop2 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼𝐵𝐸 ⊆ 𝒫 𝑉))

Proof of Theorem subgrprop2
StepHypRef Expression
1 issubgr.v . . 3 𝑉 = (Vtx‘𝑆)
2 issubgr.a . . 3 𝐴 = (Vtx‘𝐺)
3 issubgr.i . . 3 𝐼 = (iEdg‘𝑆)
4 issubgr.b . . 3 𝐵 = (iEdg‘𝐺)
5 issubgr.e . . 3 𝐸 = (Edg‘𝑆)
61, 2, 3, 4, 5subgrprop 29290 . 2 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))
7 resss 6019 . . . 4 (𝐵 ↾ dom 𝐼) ⊆ 𝐵
8 sseq1 4009 . . . 4 (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵))
97, 8mpbiri 258 . . 3 (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼𝐵)
1093anim2i 1154 . 2 ((𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉) → (𝑉𝐴𝐼𝐵𝐸 ⊆ 𝒫 𝑉))
116, 10syl 17 1 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼𝐵𝐸 ⊆ 𝒫 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wss 3951  𝒫 cpw 4600   class class class wbr 5143  dom cdm 5685  cres 5687  cfv 6561  Vtxcvtx 29013  iEdgciedg 29014  Edgcedg 29064   SubGraph csubgr 29284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-dm 5695  df-res 5697  df-iota 6514  df-fv 6569  df-subgr 29285
This theorem is referenced by:  uhgrissubgr  29292  subgrprop3  29293  subgrfun  29298  subgreldmiedg  29300  subgruhgredgd  29301  subumgredg2  29302  subuhgr  29303  subupgr  29304  subumgr  29305  subusgr  29306  subgrwlk  35137
  Copyright terms: Public domain W3C validator