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Theorem uhgrvtxedgiedgb 26929
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 26842 . . . . . . 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 5771 . . . . . 6 ran 𝐼 = ran (iEdg‘𝐺)
62, 3, 53eqtr4g 2858 . . . . 5 (𝐺 ∈ UHGraph → 𝐸 = ran 𝐼)
76rexeqdv 3365 . . . 4 (𝐺 ∈ UHGraph → (∃𝑒𝐸 𝑈𝑒 ↔ ∃𝑒 ∈ ran 𝐼 𝑈𝑒))
84uhgrfun 26859 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐼)
98funfnd 6355 . . . . 5 (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼)
10 eleq2 2878 . . . . . 6 (𝑒 = (𝐼𝑖) → (𝑈𝑒𝑈 ∈ (𝐼𝑖)))
1110rexrn 6830 . . . . 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 282 . . 3 (𝐺 ∈ UHGraph → (∃𝑒𝐸 𝑈𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖)))
1413adantr 484 . 2 ((𝐺 ∈ UHGraph ∧ 𝑈𝑉) → (∃𝑒𝐸 𝑈𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖)))
1514bicomd 226 1 ((𝐺 ∈ UHGraph ∧ 𝑈𝑉) → (∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖) ↔ ∃𝑒𝐸 𝑈𝑒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  wrex 3107  dom cdm 5519  ran crn 5520   Fn wfn 6319  cfv 6324  iEdgciedg 26790  Edgcedg 26840  UHGraphcuhgr 26849
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-edg 26841  df-uhgr 26851
This theorem is referenced by:  vtxduhgr0edgnel  27284
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