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Theorem subgrprop2 40498
 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 40497 . 2 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))
7 resss 5342 . . . 4 (𝐵 ↾ dom 𝐼) ⊆ 𝐵
8 sseq1 3589 . . . 4 (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵))
97, 8mpbiri 247 . . 3 (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼𝐵)
1093anim2i 1242 . 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 1031   = wceq 1475   ⊆ wss 3540  𝒫 cpw 4108   class class class wbr 4583  dom cdm 5038   ↾ cres 5040  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674  Edgcedga 25792   SubGraph csubgr 40491 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-dm 5048  df-res 5050  df-iota 5768  df-fv 5812  df-subgr 40492 This theorem is referenced by:  uhgrissubgr  40499  subgrprop3  40500  subgrfun  40505  subgreldmiedg  40507  subgruhgredgd  40508  subumgredg2  40509  subuhgr  40510  subupgr  40511  subumgr  40512  subusgr  40513
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