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Theorem uhgrvtxedgiedgbOLD 26435
 Description: Obsolete version of uhgrvtxedgiedgb 26434 as of 6-Jul-2022. (Contributed by AV, 24-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
uhgrvtxedgiedgbOLD.v 𝑉 = (Vtx‘𝐺)
uhgrvtxedgiedgbOLD.i 𝐼 = (iEdg‘𝐺)
uhgrvtxedgiedgbOLD.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
uhgrvtxedgiedgbOLD ((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 uhgrvtxedgiedgbOLD
StepHypRef Expression
1 edgval 26347 . . . . . . 7 (Edg‘𝐺) = ran (iEdg‘𝐺)
21a1i 11 . . . . . 6 (𝐺 ∈ UHGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
3 uhgrvtxedgiedgbOLD.e . . . . . 6 𝐸 = (Edg‘𝐺)
4 uhgrvtxedgiedgbOLD.i . . . . . . 7 𝐼 = (iEdg‘𝐺)
54rneqi 5584 . . . . . 6 ran 𝐼 = ran (iEdg‘𝐺)
62, 3, 53eqtr4g 2886 . . . . 5 (𝐺 ∈ UHGraph → 𝐸 = ran 𝐼)
76rexeqdv 3357 . . . 4 (𝐺 ∈ UHGraph → (∃𝑒𝐸 𝑈𝑒 ↔ ∃𝑒 ∈ ran 𝐼 𝑈𝑒))
84uhgrfun 26364 . . . . . 6 (𝐺 ∈ UHGraph → Fun 𝐼)
9 funfn 6153 . . . . . 6 (Fun 𝐼𝐼 Fn dom 𝐼)
108, 9sylib 210 . . . . 5 (𝐺 ∈ UHGraph → 𝐼 Fn dom 𝐼)
11 eleq2 2895 . . . . . 6 (𝑒 = (𝐼𝑖) → (𝑈𝑒𝑈 ∈ (𝐼𝑖)))
1211rexrn 6610 . . . . 5 (𝐼 Fn dom 𝐼 → (∃𝑒 ∈ ran 𝐼 𝑈𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖)))
1310, 12syl 17 . . . 4 (𝐺 ∈ UHGraph → (∃𝑒 ∈ ran 𝐼 𝑈𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖)))
147, 13bitrd 271 . . 3 (𝐺 ∈ UHGraph → (∃𝑒𝐸 𝑈𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖)))
1514adantr 474 . 2 ((𝐺 ∈ UHGraph ∧ 𝑈𝑉) → (∃𝑒𝐸 𝑈𝑒 ↔ ∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖)))
1615bicomd 215 1 ((𝐺 ∈ UHGraph ∧ 𝑈𝑉) → (∃𝑖 ∈ dom 𝐼 𝑈 ∈ (𝐼𝑖) ↔ ∃𝑒𝐸 𝑈𝑒))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  ∃wrex 3118  dom cdm 5342  ran crn 5343  Fun wfun 6117   Fn wfn 6118  ‘cfv 6123  Vtxcvtx 26294  iEdgciedg 26295  Edgcedg 26345  UHGraphcuhgr 26354 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-fv 6131  df-edg 26346  df-uhgr 26356 This theorem is referenced by: (None)
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