MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  subgrprop3 Structured version   Visualization version   GIF version

Theorem subgrprop3 29131
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. (Contributed by AV, 19-Nov-2020.)
Hypotheses
Ref Expression
subgrprop3.v 𝑉 = (Vtx‘𝑆)
subgrprop3.a 𝐴 = (Vtx‘𝐺)
subgrprop3.e 𝐸 = (Edg‘𝑆)
subgrprop3.b 𝐵 = (Edg‘𝐺)
Assertion
Ref Expression
subgrprop3 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐸𝐵))

Proof of Theorem subgrprop3
StepHypRef Expression
1 subgrprop3.v . . . 4 𝑉 = (Vtx‘𝑆)
2 subgrprop3.a . . . 4 𝐴 = (Vtx‘𝐺)
3 eqid 2725 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
4 eqid 2725 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
5 subgrprop3.e . . . 4 𝐸 = (Edg‘𝑆)
61, 2, 3, 4, 5subgrprop2 29129 . . 3 (𝑆 SubGraph 𝐺 → (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝐸 ⊆ 𝒫 𝑉))
7 3simpa 1145 . . 3 ((𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝐸 ⊆ 𝒫 𝑉) → (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)))
86, 7syl 17 . 2 (𝑆 SubGraph 𝐺 → (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺)))
9 simprl 769 . . 3 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → 𝑉𝐴)
10 rnss 5935 . . . . 5 ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
1110ad2antll 727 . . . 4 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
12 subgrv 29125 . . . . . 6 (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
13 edgval 28904 . . . . . . . . 9 (Edg‘𝑆) = ran (iEdg‘𝑆)
1413a1i 11 . . . . . . . 8 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝑆) = ran (iEdg‘𝑆))
155, 14eqtrid 2777 . . . . . . 7 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐸 = ran (iEdg‘𝑆))
16 subgrprop3.b . . . . . . . 8 𝐵 = (Edg‘𝐺)
17 edgval 28904 . . . . . . . . 9 (Edg‘𝐺) = ran (iEdg‘𝐺)
1817a1i 11 . . . . . . . 8 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝐺) = ran (iEdg‘𝐺))
1916, 18eqtrid 2777 . . . . . . 7 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐵 = ran (iEdg‘𝐺))
2015, 19sseq12d 4006 . . . . . 6 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (𝐸𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
2112, 20syl 17 . . . . 5 (𝑆 SubGraph 𝐺 → (𝐸𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
2221adantr 479 . . . 4 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → (𝐸𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
2311, 22mpbird 256 . . 3 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → 𝐸𝐵)
249, 23jca 510 . 2 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → (𝑉𝐴𝐸𝐵))
258, 24mpdan 685 1 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐸𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wcel 2098  Vcvv 3463  wss 3940  𝒫 cpw 4598   class class class wbr 5143  ran crn 5673  cfv 6542  Vtxcvtx 28851  iEdgciedg 28852  Edgcedg 28902   SubGraph csubgr 29122
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7737
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6494  df-fun 6544  df-fv 6550  df-edg 28903  df-subgr 29123
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator