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Theorem subgrprop2 26211
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 26210 . 2 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐼 = (𝐵 ↾ dom 𝐼) ∧ 𝐸 ⊆ 𝒫 𝑉))
7 resss 5457 . . . 4 (𝐵 ↾ dom 𝐼) ⊆ 𝐵
8 sseq1 3659 . . . 4 (𝐼 = (𝐵 ↾ dom 𝐼) → (𝐼𝐵 ↔ (𝐵 ↾ dom 𝐼) ⊆ 𝐵))
97, 8mpbiri 248 . . 3 (𝐼 = (𝐵 ↾ dom 𝐼) → 𝐼𝐵)
1093anim2i 1268 . 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 1054   = wceq 1523  wss 3607  𝒫 cpw 4191   class class class wbr 4685  dom cdm 5143  cres 5145  cfv 5926  Vtxcvtx 25919  iEdgciedg 25920  Edgcedg 25984   SubGraph csubgr 26204
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-dm 5153  df-res 5155  df-iota 5889  df-fv 5934  df-subgr 26205
This theorem is referenced by:  uhgrissubgr  26212  subgrprop3  26213  subgrfun  26218  subgreldmiedg  26220  subgruhgredgd  26221  subumgredg2  26222  subuhgr  26223  subupgr  26224  subumgr  26225  subusgr  26226
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