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

Theorem uhgrsubgrself 26371
Description: A hypergraph is a subgraph of itself. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)
Assertion
Ref Expression
uhgrsubgrself (𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺)

Proof of Theorem uhgrsubgrself
StepHypRef Expression
1 ssid 3765 . . 3 (Vtx‘𝐺) ⊆ (Vtx‘𝐺)
2 ssid 3765 . . 3 (iEdg‘𝐺) ⊆ (iEdg‘𝐺)
31, 2pm3.2i 470 . 2 ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))
4 eqid 2760 . . . 4 (iEdg‘𝐺) = (iEdg‘𝐺)
54uhgrfun 26160 . . 3 (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
6 id 22 . . 3 (𝐺 ∈ UHGraph → 𝐺 ∈ UHGraph)
7 eqid 2760 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
87, 7, 4, 4uhgrissubgr 26366 . . 3 ((𝐺 ∈ UHGraph ∧ Fun (iEdg‘𝐺) ∧ 𝐺 ∈ UHGraph) → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))))
95, 6, 8mpd3an23 1575 . 2 (𝐺 ∈ UHGraph → (𝐺 SubGraph 𝐺 ↔ ((Vtx‘𝐺) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝐺) ⊆ (iEdg‘𝐺))))
103, 9mpbiri 248 1 (𝐺 ∈ UHGraph → 𝐺 SubGraph 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2139  wss 3715   class class class wbr 4804  Fun wfun 6043  cfv 6049  Vtxcvtx 26073  iEdgciedg 26074  UHGraphcuhgr 26150   SubGraph csubgr 26358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fv 6057  df-edg 26139  df-uhgr 26152  df-subgr 26359
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator