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Theorem uhgrvtxedgiedgb 28936
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 28849 . . . . . . 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 5933 . . . . . 6 ran 𝐼 = ran (iEdg‘𝐺)
62, 3, 53eqtr4g 2792 . . . . 5 (𝐺 ∈ UHGraph → 𝐸 = ran 𝐼)
76rexeqdv 3321 . . . 4 (𝐺 ∈ UHGraph → (∃𝑒𝐸 𝑈𝑒 ↔ ∃𝑒 ∈ ran 𝐼 𝑈𝑒))
84uhgrfun 28866 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐼)
98funfnd 6578 . . . . 5 (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼)
10 eleq2 2817 . . . . . 6 (𝑒 = (𝐼𝑖) → (𝑈𝑒𝑈 ∈ (𝐼𝑖)))
1110rexrn 7091 . . . . 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 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 395   = wceq 1534  wcel 2099  wrex 3065  dom cdm 5672  ran crn 5673   Fn wfn 6537  cfv 6542  iEdgciedg 28797  Edgcedg 28847  UHGraphcuhgr 28856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-edg 28848  df-uhgr 28858
This theorem is referenced by:  vtxduhgr0edgnel  29295
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