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Theorem upgredg 25940
 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 25854 . . . . . 6 (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
31, 2syl5eq 2667 . . . . 5 (𝐺 ∈ UPGraph → 𝐸 = ran (iEdg‘𝐺))
43eleq2d 2684 . . . 4 (𝐺 ∈ UPGraph → (𝐶𝐸𝐶 ∈ ran (iEdg‘𝐺)))
5 upgredg.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
6 eqid 2621 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
75, 6upgrf 25890 . . . . . 6 (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
8 frn 6015 . . . . . 6 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
97, 8syl 17 . . . . 5 (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
109sseld 3586 . . . 4 (𝐺 ∈ UPGraph → (𝐶 ∈ ran (iEdg‘𝐺) → 𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
114, 10sylbid 230 . . 3 (𝐺 ∈ UPGraph → (𝐶𝐸𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
1211imp 445 . 2 ((𝐺 ∈ UPGraph ∧ 𝐶𝐸) → 𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
13 fveq2 6153 . . . . 5 (𝑥 = 𝐶 → (#‘𝑥) = (#‘𝐶))
1413breq1d 4628 . . . 4 (𝑥 = 𝐶 → ((#‘𝑥) ≤ 2 ↔ (#‘𝐶) ≤ 2))
1514elrab 3350 . . 3 (𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ≤ 2))
16 hashle2prv 13205 . . . 4 (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → ((#‘𝐶) ≤ 2 ↔ ∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏}))
1716biimpa 501 . . 3 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ≤ 2) → ∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏})
1815, 17sylbi 207 . 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 384   = wceq 1480   ∈ wcel 1987  ∃wrex 2908  {crab 2911   ∖ cdif 3556   ⊆ wss 3559  ∅c0 3896  𝒫 cpw 4135  {csn 4153  {cpr 4155   class class class wbr 4618  dom cdm 5079  ran crn 5080  ⟶wf 5848  ‘cfv 5852   ≤ cle 10026  2c2 11021  #chash 13064  Vtxcvtx 25787  iEdgciedg 25788  Edgcedg 25852   UPGraph cupgr 25884 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909  ax-cnex 9943  ax-resscn 9944  ax-1cn 9945  ax-icn 9946  ax-addcl 9947  ax-addrcl 9948  ax-mulcl 9949  ax-mulrcl 9950  ax-mulcom 9951  ax-addass 9952  ax-mulass 9953  ax-distr 9954  ax-i2m1 9955  ax-1ne0 9956  ax-1rid 9957  ax-rnegex 9958  ax-rrecex 9959  ax-cnre 9960  ax-pre-lttri 9961  ax-pre-lttrn 9962  ax-pre-ltadd 9963  ax-pre-mulgt0 9964 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-pred 5644  df-ord 5690  df-on 5691  df-lim 5692  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-riota 6571  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-om 7020  df-1st 7120  df-2nd 7121  df-wrecs 7359  df-recs 7420  df-rdg 7458  df-1o 7512  df-2o 7513  df-oadd 7516  df-er 7694  df-en 7907  df-dom 7908  df-sdom 7909  df-fin 7910  df-card 8716  df-cda 8941  df-pnf 10027  df-mnf 10028  df-xr 10029  df-ltxr 10030  df-le 10031  df-sub 10219  df-neg 10220  df-nn 10972  df-2 11030  df-n0 11244  df-xnn0 11315  df-z 11329  df-uz 11639  df-fz 12276  df-hash 13065  df-edg 25853  df-upgr 25886 This theorem is referenced by:  upgrpredgv  25942  upgredg2vtx  25944  upgredgpr  25945
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