MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uhgrn0 Structured version   Visualization version   GIF version

Theorem uhgrn0 25956
Description: An edge is a nonempty subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 15-Dec-2020.)
Hypothesis
Ref Expression
uhgrfun.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
uhgrn0 ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ≠ ∅)

Proof of Theorem uhgrn0
StepHypRef Expression
1 eqid 2621 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
2 uhgrfun.e . . . . . . 7 𝐸 = (iEdg‘𝐺)
31, 2uhgrf 25951 . . . . . 6 (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4 fndm 5988 . . . . . . 7 (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
54feq2d 6029 . . . . . 6 (𝐸 Fn 𝐴 → (𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
63, 5syl5ibcom 235 . . . . 5 (𝐺 ∈ UHGraph → (𝐸 Fn 𝐴𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
76imp 445 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
87ffvelrnda 6357 . . 3 (((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) ∧ 𝐹𝐴) → (𝐸𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
983impa 1258 . 2 ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
10 eldifsni 4318 . 2 ((𝐸𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → (𝐸𝐹) ≠ ∅)
119, 10syl 17 1 ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1037   = wceq 1482  wcel 1989  wne 2793  cdif 3569  c0 3913  𝒫 cpw 4156  {csn 4175  dom cdm 5112   Fn wfn 5881  wf 5882  cfv 5886  Vtxcvtx 25868  iEdgciedg 25869   UHGraph cuhgr 25945
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894  df-uhgr 25947
This theorem is referenced by:  lpvtx  25957  subgruhgredgd  26170
  Copyright terms: Public domain W3C validator