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Theorem subgrprop2 29306
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 29305 . 2 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))
7 resss 6022 . . . 4 (𝐵 ↾ dom 𝐼) ⊆ 𝐵
8 sseq1 4021 . . . 4 (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵))
97, 8mpbiri 258 . . 3 (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼𝐵)
1093anim2i 1152 . 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 1086   = wceq 1537  wss 3963  𝒫 cpw 4605   class class class wbr 5148  dom cdm 5689  cres 5691  cfv 6563  Vtxcvtx 29028  iEdgciedg 29029  Edgcedg 29079   SubGraph csubgr 29299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-dm 5699  df-res 5701  df-iota 6516  df-fv 6571  df-subgr 29300
This theorem is referenced by:  uhgrissubgr  29307  subgrprop3  29308  subgrfun  29313  subgreldmiedg  29315  subgruhgredgd  29316  subumgredg2  29317  subuhgr  29318  subupgr  29319  subumgr  29320  subusgr  29321  subgrwlk  35117
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