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Theorem subgrprop 29209
Description: The properties of a subgraph. (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
subgrprop (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))

Proof of Theorem subgrprop
StepHypRef Expression
1 subgrv 29206 . 2 (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
2 issubgr.v . . . . 5 𝑉 = (Vtx‘𝑆)
3 issubgr.a . . . . 5 𝐴 = (Vtx‘𝐺)
4 issubgr.i . . . . 5 𝐼 = (iEdg‘𝑆)
5 issubgr.b . . . . 5 𝐵 = (iEdg‘𝐺)
6 issubgr.e . . . . 5 𝐸 = (Edg‘𝑆)
72, 3, 4, 5, 6issubgr 29207 . . . 4 ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝑆 SubGraph 𝐺 ↔ (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
87biimpd 228 . . 3 ((𝐺 ∈ V ∧ 𝑆 ∈ V) → (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
98ancoms 457 . 2 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉)))
101, 9mpcom 38 1 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084   = wceq 1534  wcel 2099  Vcvv 3462  wss 3947  𝒫 cpw 4607   class class class wbr 5153  dom cdm 5682  cres 5684  cfv 6554  Vtxcvtx 28932  iEdgciedg 28933  Edgcedg 28983   SubGraph csubgr 29203
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-xp 5688  df-rel 5689  df-dm 5692  df-res 5694  df-iota 6506  df-fv 6562  df-subgr 29204
This theorem is referenced by:  subgrprop2  29210  subgrwlk  34960
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