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

Theorem subgrprop3 28266
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 2737 . . . 4 (iEdg‘𝑆) = (iEdg‘𝑆)
4 eqid 2737 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
5 subgrprop3.e . . . 4 𝐸 = (Edg‘𝑆)
61, 2, 3, 4, 5subgrprop2 28264 . . 3 (𝑆 SubGraph 𝐺 → (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺) ∧ 𝐸 ⊆ 𝒫 𝑉))
7 3simpa 1149 . . 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 770 . . 3 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → 𝑉𝐴)
10 rnss 5899 . . . . 5 ((iEdg‘𝑆) ⊆ (iEdg‘𝐺) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
1110ad2antll 728 . . . 4 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺))
12 subgrv 28260 . . . . . 6 (𝑆 SubGraph 𝐺 → (𝑆 ∈ V ∧ 𝐺 ∈ V))
13 edgval 28042 . . . . . . . . 9 (Edg‘𝑆) = ran (iEdg‘𝑆)
1413a1i 11 . . . . . . . 8 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝑆) = ran (iEdg‘𝑆))
155, 14eqtrid 2789 . . . . . . 7 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐸 = ran (iEdg‘𝑆))
16 subgrprop3.b . . . . . . . 8 𝐵 = (Edg‘𝐺)
17 edgval 28042 . . . . . . . . 9 (Edg‘𝐺) = ran (iEdg‘𝐺)
1817a1i 11 . . . . . . . 8 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → (Edg‘𝐺) = ran (iEdg‘𝐺))
1916, 18eqtrid 2789 . . . . . . 7 ((𝑆 ∈ V ∧ 𝐺 ∈ V) → 𝐵 = ran (iEdg‘𝐺))
2015, 19sseq12d 3982 . . . . . 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 482 . . . 4 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → (𝐸𝐵 ↔ ran (iEdg‘𝑆) ⊆ ran (iEdg‘𝐺)))
2311, 22mpbird 257 . . 3 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → 𝐸𝐵)
249, 23jca 513 . 2 ((𝑆 SubGraph 𝐺 ∧ (𝑉𝐴 ∧ (iEdg‘𝑆) ⊆ (iEdg‘𝐺))) → (𝑉𝐴𝐸𝐵))
258, 24mpdan 686 1 (𝑆 SubGraph 𝐺 → (𝑉𝐴𝐸𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  Vcvv 3448  wss 3915  𝒫 cpw 4565   class class class wbr 5110  ran crn 5639  cfv 6501  Vtxcvtx 27989  iEdgciedg 27990  Edgcedg 28040   SubGraph csubgr 28257
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-iota 6453  df-fun 6503  df-fv 6509  df-edg 28041  df-subgr 28258
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator