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Theorem uhgrvtxedgiedgb 29153
Description: In a hypergraph, a vertex is incident with an edge iff it is contained in an element of the range of the edge function. (Contributed by AV, 24-Dec-2020.) (Revised by AV, 6-Jul-2022.)
Hypotheses
Ref Expression
uhgrvtxedgiedgb.i 𝐼 = (iEdg‘𝐺)
uhgrvtxedgiedgb.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
uhgrvtxedgiedgb ((𝐺 ∈ UHGraph ∧ 𝑈𝑉) → (∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖) ↔ ∃𝑒𝐸 𝑈𝑒))
Distinct variable groups:   𝑒,𝐸   𝑒,𝐼,𝑖   𝑈,𝑒,𝑖
Allowed substitution hints:   𝐸(𝑖)   𝐺(𝑒,𝑖)   𝑉(𝑒,𝑖)

Proof of Theorem uhgrvtxedgiedgb
StepHypRef Expression
1 edgval 29066 . . . . . . 7 (Edg‘𝐺) = ran (iEdg‘𝐺)
21a1i 11 . . . . . 6 (𝐺 ∈ UHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
3 uhgrvtxedgiedgb.e . . . . . 6 𝐸 = (Edg‘𝐺)
4 uhgrvtxedgiedgb.i . . . . . . 7 𝐼 = (iEdg‘𝐺)
54rneqi 5948 . . . . . 6 ran 𝐼 = ran (iEdg‘𝐺)
62, 3, 53eqtr4g 2802 . . . . 5 (𝐺 ∈ UHGraph → 𝐸 = ran 𝐼)
76rexeqdv 3327 . . . 4 (𝐺 ∈ UHGraph → (∃𝑒𝐸 𝑈𝑒 ↔ ∃𝑒 ∈ ran 𝐼 𝑈𝑒))
84uhgrfun 29083 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐼)
98funfnd 6597 . . . . 5 (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼)
10 eleq2 2830 . . . . . 6 (𝑒 = (𝐼𝑖) → (𝑈𝑒𝑈 ∈ (𝐼𝑖)))
1110rexrn 7107 . . . . 5 (𝐼 Fn dom 𝐼 → (∃𝑒 ∈ ran 𝐼 𝑈𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖)))
129, 11syl 17 . . . 4 (𝐺 ∈ UHGraph → (∃𝑒 ∈ ran 𝐼 𝑈𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖)))
137, 12bitrd 279 . . 3 (𝐺 ∈ UHGraph → (∃𝑒𝐸 𝑈𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖)))
1413adantr 480 . 2 ((𝐺 ∈ UHGraph ∧ 𝑈𝑉) → (∃𝑒𝐸 𝑈𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖)))
1514bicomd 223 1 ((𝐺 ∈ UHGraph ∧ 𝑈𝑉) → (∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖) ↔ ∃𝑒𝐸 𝑈𝑒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wrex 3070  dom cdm 5685  ran crn 5686   Fn wfn 6556  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:  vtxduhgr0edgnel  29512
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