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Theorem uhgrn0 26183
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 2761 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
2 uhgrfun.e . . . . . . 7 𝐸 = (iEdg‘𝐺)
31, 2uhgrf 26178 . . . . . 6 (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4 fndm 6152 . . . . . . 7 (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
54feq2d 6193 . . . . . 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 444 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
87ffvelrnda 6524 . . 3 (((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) ∧ 𝐹𝐴) → (𝐸𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
983impa 1101 . 2 ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
10 eldifsni 4467 . 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 383  w3a 1072   = wceq 1632  wcel 2140  wne 2933  cdif 3713  c0 4059  𝒫 cpw 4303  {csn 4322  dom cdm 5267   Fn wfn 6045  wf 6046  cfv 6050  Vtxcvtx 26095  iEdgciedg 26096  UHGraphcuhgr 26172
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 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056
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 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-id 5175  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-fv 6058  df-uhgr 26174
This theorem is referenced by:  lpvtx  26184  subgruhgredgd  26397
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