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Theorem upgredg 26928
 Description: For each edge in a pseudograph, there are two vertices which are connected by this edge. (Contributed by AV, 4-Nov-2020.) (Proof shortened by AV, 26-Nov-2021.)
Hypotheses
Ref Expression
upgredg.v 𝑉 = (Vtx‘𝐺)
upgredg.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
upgredg ((𝐺 ∈ UPGraph ∧ 𝐶𝐸) → ∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏})
Distinct variable groups:   𝐶,𝑎,𝑏   𝐺,𝑎,𝑏   𝑉,𝑎,𝑏
Allowed substitution hints:   𝐸(𝑎,𝑏)

Proof of Theorem upgredg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 upgredg.e . . . . . 6 𝐸 = (Edg‘𝐺)
2 edgval 26840 . . . . . . 7 (Edg‘𝐺) = ran (iEdg‘𝐺)
32a1i 11 . . . . . 6 (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
41, 3syl5eq 2869 . . . . 5 (𝐺 ∈ UPGraph → 𝐸 = ran (iEdg‘𝐺))
54eleq2d 2899 . . . 4 (𝐺 ∈ UPGraph → (𝐶𝐸𝐶 ∈ ran (iEdg‘𝐺)))
6 upgredg.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
7 eqid 2822 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
86, 7upgrf 26877 . . . . . 6 (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
98frnd 6501 . . . . 5 (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
109sseld 3941 . . . 4 (𝐺 ∈ UPGraph → (𝐶 ∈ ran (iEdg‘𝐺) → 𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
115, 10sylbid 243 . . 3 (𝐺 ∈ UPGraph → (𝐶𝐸𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
1211imp 410 . 2 ((𝐺 ∈ UPGraph ∧ 𝐶𝐸) → 𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
13 fveq2 6652 . . . . 5 (𝑥 = 𝐶 → (♯‘𝑥) = (♯‘𝐶))
1413breq1d 5052 . . . 4 (𝑥 = 𝐶 → ((♯‘𝑥) ≤ 2 ↔ (♯‘𝐶) ≤ 2))
1514elrab 3655 . . 3 (𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘𝐶) ≤ 2))
16 hashle2prv 13832 . . . 4 (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → ((♯‘𝐶) ≤ 2 ↔ ∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏}))
1716biimpa 480 . . 3 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘𝐶) ≤ 2) → ∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏})
1815, 17sylbi 220 . 2 (𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → ∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏})
1912, 18syl 17 1 ((𝐺 ∈ UPGraph ∧ 𝐶𝐸) → ∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2114  ∃wrex 3131  {crab 3134   ∖ cdif 3905  ∅c0 4265  𝒫 cpw 4511  {csn 4539  {cpr 4541   class class class wbr 5042  dom cdm 5532  ran crn 5533  ‘cfv 6334   ≤ cle 10665  2c2 11680  ♯chash 13686  Vtxcvtx 26787  iEdgciedg 26788  Edgcedg 26838  UPGraphcupgr 26871 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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rmo 3138  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-2o 8090  df-oadd 8093  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-fin 8500  df-dju 9318  df-card 9356  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12886  df-hash 13687  df-edg 26839  df-upgr 26873 This theorem is referenced by:  upgrpredgv  26930  upgredg2vtx  26932  upgredgpr  26933  edglnl  26934  numedglnl  26935  isomuspgrlem1  44284  isomuspgrlem2b  44286  isomuspgrlem2d  44288
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