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Theorem uhgredgiedgb 29143
Description: In a hypergraph, a set is an edge iff it is an indexed edge. (Contributed by AV, 17-Oct-2020.)
Hypothesis
Ref Expression
uhgredgiedgb.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
uhgredgiedgb (𝐺 ∈ UHGraph → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))
Distinct variable groups:   𝑥,𝐸   𝑥,𝐼
Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem uhgredgiedgb
StepHypRef Expression
1 uhgredgiedgb.i . . 3 𝐼 = (iEdg‘𝐺)
21uhgrfun 29083 . 2 (𝐺 ∈ UHGraph → Fun 𝐼)
31edgiedgb 29071 . 2 (Fun 𝐼 → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))
42, 3syl 17 1 (𝐺 ∈ UHGraph → (𝐸 ∈ (Edg‘𝐺) ↔ ∃𝑥 ∈ dom 𝐼 𝐸 = (𝐼𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108  wrex 3070  dom cdm 5685  Fun wfun 6555  cfv 6561  iEdgciedg 29014  Edgcedg 29064  UHGraphcuhgr 29073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-edg 29065  df-uhgr 29075
This theorem is referenced by:  usgredg2vtxeuALT  29239  vtxduhgr0nedg  29510  umgr2wlk  29969  1pthon2v  30172  uhgr3cyclex  30201  isuspgrim0  47872  clnbgrgrimlem  47901  clnbgrgrim  47902  grimedg  47903
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