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Theorem uhgrn0 27024
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 2739 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
2 uhgrfun.e . . . . . . 7 𝐸 = (iEdg‘𝐺)
31, 2uhgrf 27019 . . . . . 6 (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
4 fndm 6450 . . . . . . 7 (𝐸 Fn 𝐴 → dom 𝐸 = 𝐴)
54feq2d 6500 . . . . . 6 (𝐸 Fn 𝐴 → (𝐸:dom 𝐸⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
63, 5syl5ibcom 248 . . . . 5 (𝐺 ∈ UHGraph → (𝐸 Fn 𝐴𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅})))
76imp 410 . . . 4 ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
87ffvelrnda 6873 . . 3 (((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴) ∧ 𝐹𝐴) → (𝐸𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
983impa 1111 . 2 ((𝐺 ∈ UHGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
10 eldifsni 4688 . 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 399  w3a 1088   = wceq 1542  wcel 2114  wne 2935  cdif 3850  c0 4221  𝒫 cpw 4498  {csn 4526  dom cdm 5535   Fn wfn 6344  wf 6345  cfv 6349  Vtxcvtx 26953  iEdgciedg 26954  UHGraphcuhgr 27013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-pw 4500  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-rn 5546  df-iota 6307  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357  df-uhgr 27015
This theorem is referenced by:  lpvtx  27025  subgruhgredgd  27238
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