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Theorem subgrprop2 27070
 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 27069 . 2 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))
7 resss 5865 . . . 4 (𝐵 ↾ dom 𝐼) ⊆ 𝐵
8 sseq1 3978 . . . 4 (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵))
97, 8mpbiri 261 . . 3 (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼𝐵)
1093anim2i 1150 . 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 1084   = wceq 1538   ⊆ wss 3919  𝒫 cpw 4522   class class class wbr 5052  dom cdm 5542   ↾ cres 5544  ‘cfv 6343  Vtxcvtx 26795  iEdgciedg 26796  Edgcedg 26846   SubGraph csubgr 27063 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-xp 5548  df-rel 5549  df-dm 5552  df-res 5554  df-iota 6302  df-fv 6351  df-subgr 27064 This theorem is referenced by:  uhgrissubgr  27071  subgrprop3  27072  subgrfun  27077  subgreldmiedg  27079  subgruhgredgd  27080  subumgredg2  27081  subuhgr  27082  subupgr  27083  subumgr  27084  subusgr  27085  subgrwlk  32439
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