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Theorem uhgrn0 27437
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 2738 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
2 uhgrfun.e . . . . . . 7 𝐸 = (iEdg‘𝐺)
31, 2uhgrf 27432 . . . . . 6 (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4 fndm 6536 . . . . . . 7 (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
54feq2d 6586 . . . . . 6 (𝐸 Fn 𝐴 → (𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
63, 5syl5ibcom 244 . . . . 5 (𝐺 ∈ UHGraph → (𝐸 Fn 𝐴𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
76imp 407 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
87ffvelrnda 6961 . . 3 (((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) ∧ 𝐹𝐴) → (𝐸𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
983impa 1109 . 2 ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
10 eldifsni 4723 . 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 1086   = wceq 1539  wcel 2106  wne 2943  cdif 3884  c0 4256  𝒫 cpw 4533  {csn 4561  dom cdm 5589   Fn wfn 6428  wf 6429  cfv 6433  Vtxcvtx 27366  iEdgciedg 27367  UHGraphcuhgr 27426
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-uhgr 27428
This theorem is referenced by:  lpvtx  27438  subgruhgredgd  27651
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