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Theorem uhgrvtxedgiedgb 28385
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 28298 . . . . . . 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 5934 . . . . . 6 ran 𝐼 = ran (iEdg‘𝐺)
62, 3, 53eqtr4g 2797 . . . . 5 (𝐺 ∈ UHGraph → 𝐸 = ran 𝐼)
76rexeqdv 3326 . . . 4 (𝐺 ∈ UHGraph → (∃𝑒𝐸 𝑈𝑒 ↔ ∃𝑒 ∈ ran 𝐼 𝑈𝑒))
84uhgrfun 28315 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐼)
98funfnd 6576 . . . . 5 (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼)
10 eleq2 2822 . . . . . 6 (𝑒 = (𝐼𝑖) → (𝑈𝑒𝑈 ∈ (𝐼𝑖)))
1110rexrn 7085 . . . . 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 278 . . 3 (𝐺 ∈ UHGraph → (∃𝑒𝐸 𝑈𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖)))
1413adantr 481 . 2 ((𝐺 ∈ UHGraph ∧ 𝑈𝑉) → (∃𝑒𝐸 𝑈𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖)))
1514bicomd 222 1 ((𝐺 ∈ UHGraph ∧ 𝑈𝑉) → (∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖) ↔ ∃𝑒𝐸 𝑈𝑒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  wrex 3070  dom cdm 5675  ran crn 5676   Fn wfn 6535  cfv 6540  iEdgciedg 28246  Edgcedg 28296  UHGraphcuhgr 28305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-edg 28297  df-uhgr 28307
This theorem is referenced by:  vtxduhgr0edgnel  28740
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