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Theorem uhgrn0 26785
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 2826 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
2 uhgrfun.e . . . . . . 7 𝐸 = (iEdg‘𝐺)
31, 2uhgrf 26780 . . . . . 6 (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4 fndm 6454 . . . . . . 7 (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
54feq2d 6499 . . . . . 6 (𝐸 Fn 𝐴 → (𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
63, 5syl5ibcom 246 . . . . 5 (𝐺 ∈ UHGraph → (𝐸 Fn 𝐴𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
76imp 407 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
87ffvelrnda 6849 . . 3 (((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) ∧ 𝐹𝐴) → (𝐸𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
983impa 1104 . 2 ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
10 eldifsni 4721 . 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 396  w3a 1081   = wceq 1530  wcel 2107  wne 3021  cdif 3937  c0 4295  𝒫 cpw 4542  {csn 4564  dom cdm 5554   Fn wfn 6349  wf 6350  cfv 6354  Vtxcvtx 26714  iEdgciedg 26715  UHGraphcuhgr 26774
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362  df-uhgr 26776
This theorem is referenced by:  lpvtx  26786  subgruhgredgd  26999
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