Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  upgredg Structured version   Visualization version   GIF version

Theorem upgredg 25811
 Description: For each edge in a pseudograph, there are two vertices which are connected by this edge. (Contributed by AV, 4-Nov-2020.)
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 variables 𝑥 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 upgredg.e . . . . . . 7 𝐸 = (Edg‘𝐺)
2 edgaval 25794 . . . . . . 7 (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
31, 2syl5eq 2656 . . . . . 6 (𝐺 ∈ UPGraph → 𝐸 = ran (iEdg‘𝐺))
43eleq2d 2673 . . . . 5 (𝐺 ∈ UPGraph → (𝐶𝐸𝐶 ∈ ran (iEdg‘𝐺)))
5 upgredg.v . . . . . . . 8 𝑉 = (Vtx‘𝐺)
6 eqid 2610 . . . . . . . 8 (iEdg‘𝐺) = (iEdg‘𝐺)
75, 6upgrf 25753 . . . . . . 7 (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
8 frn 5966 . . . . . . 7 ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
97, 8syl 17 . . . . . 6 (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
109sseld 3567 . . . . 5 (𝐺 ∈ UPGraph → (𝐶 ∈ ran (iEdg‘𝐺) → 𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
114, 10sylbid 229 . . . 4 (𝐺 ∈ UPGraph → (𝐶𝐸𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
1211imp 444 . . 3 ((𝐺 ∈ UPGraph ∧ 𝐶𝐸) → 𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
13 fveq2 6103 . . . . . 6 (𝑥 = 𝐶 → (#‘𝑥) = (#‘𝐶))
1413breq1d 4593 . . . . 5 (𝑥 = 𝐶 → ((#‘𝑥) ≤ 2 ↔ (#‘𝐶) ≤ 2))
1514elrab 3331 . . . 4 (𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ≤ 2))
16 2nn0 11186 . . . . . . 7 2 ∈ ℕ0
17 hashbnd 12985 . . . . . . 7 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 2 ∈ ℕ0 ∧ (#‘𝐶) ≤ 2) → 𝐶 ∈ Fin)
1816, 17mp3an2 1404 . . . . . 6 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ≤ 2) → 𝐶 ∈ Fin)
19 hashcl 13009 . . . . . 6 (𝐶 ∈ Fin → (#‘𝐶) ∈ ℕ0)
2018, 19syl 17 . . . . 5 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ≤ 2) → (#‘𝐶) ∈ ℕ0)
21 nn0re 11178 . . . . . . . . 9 ((#‘𝐶) ∈ ℕ0 → (#‘𝐶) ∈ ℝ)
22 2re 10967 . . . . . . . . . 10 2 ∈ ℝ
2322a1i 11 . . . . . . . . 9 ((#‘𝐶) ∈ ℕ0 → 2 ∈ ℝ)
2421, 23leloed 10059 . . . . . . . 8 ((#‘𝐶) ∈ ℕ0 → ((#‘𝐶) ≤ 2 ↔ ((#‘𝐶) < 2 ∨ (#‘𝐶) = 2)))
2524adantl 481 . . . . . . 7 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → ((#‘𝐶) ≤ 2 ↔ ((#‘𝐶) < 2 ∨ (#‘𝐶) = 2)))
26 nn0lt2 11317 . . . . . . . . . . 11 (((#‘𝐶) ∈ ℕ0 ∧ (#‘𝐶) < 2) → ((#‘𝐶) = 0 ∨ (#‘𝐶) = 1))
2726ex 449 . . . . . . . . . 10 ((#‘𝐶) ∈ ℕ0 → ((#‘𝐶) < 2 → ((#‘𝐶) = 0 ∨ (#‘𝐶) = 1)))
2827adantl 481 . . . . . . . . 9 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → ((#‘𝐶) < 2 → ((#‘𝐶) = 0 ∨ (#‘𝐶) = 1)))
29 eldifsn 4260 . . . . . . . . . . . 12 (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ↔ (𝐶 ∈ 𝒫 𝑉𝐶 ≠ ∅))
30 hasheq0 13015 . . . . . . . . . . . . . 14 (𝐶 ∈ 𝒫 𝑉 → ((#‘𝐶) = 0 ↔ 𝐶 = ∅))
3130adantr 480 . . . . . . . . . . . . 13 ((𝐶 ∈ 𝒫 𝑉𝐶 ≠ ∅) → ((#‘𝐶) = 0 ↔ 𝐶 = ∅))
32 eqneqall 2793 . . . . . . . . . . . . . . 15 (𝐶 = ∅ → (𝐶 ≠ ∅ → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
3332com12 32 . . . . . . . . . . . . . 14 (𝐶 ≠ ∅ → (𝐶 = ∅ → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
3433adantl 481 . . . . . . . . . . . . 13 ((𝐶 ∈ 𝒫 𝑉𝐶 ≠ ∅) → (𝐶 = ∅ → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
3531, 34sylbid 229 . . . . . . . . . . . 12 ((𝐶 ∈ 𝒫 𝑉𝐶 ≠ ∅) → ((#‘𝐶) = 0 → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
3629, 35sylbi 206 . . . . . . . . . . 11 (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → ((#‘𝐶) = 0 → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
3736adantr 480 . . . . . . . . . 10 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → ((#‘𝐶) = 0 → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
38 hash1snb 13068 . . . . . . . . . . . 12 (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → ((#‘𝐶) = 1 ↔ ∃𝑐 𝐶 = {𝑐}))
39 eleq1 2676 . . . . . . . . . . . . . . . . . . . . 21 (𝐶 = {𝑐} → (𝐶 ∈ 𝒫 𝑉 ↔ {𝑐} ∈ 𝒫 𝑉))
40 vex 3176 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑐 ∈ V
4140snelpw 4840 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑐𝑉 ↔ {𝑐} ∈ 𝒫 𝑉)
4241biimpri 217 . . . . . . . . . . . . . . . . . . . . . 22 ({𝑐} ∈ 𝒫 𝑉𝑐𝑉)
4342a1i 11 . . . . . . . . . . . . . . . . . . . . 21 (𝐶 = {𝑐} → ({𝑐} ∈ 𝒫 𝑉𝑐𝑉))
4439, 43sylbid 229 . . . . . . . . . . . . . . . . . . . 20 (𝐶 = {𝑐} → (𝐶 ∈ 𝒫 𝑉𝑐𝑉))
4544com12 32 . . . . . . . . . . . . . . . . . . 19 (𝐶 ∈ 𝒫 𝑉 → (𝐶 = {𝑐} → 𝑐𝑉))
4645adantr 480 . . . . . . . . . . . . . . . . . 18 ((𝐶 ∈ 𝒫 𝑉𝐶 ≠ ∅) → (𝐶 = {𝑐} → 𝑐𝑉))
4729, 46sylbi 206 . . . . . . . . . . . . . . . . 17 (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → (𝐶 = {𝑐} → 𝑐𝑉))
4847imp 444 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 𝐶 = {𝑐}) → 𝑐𝑉)
49 id 22 . . . . . . . . . . . . . . . . . 18 (𝐶 = {𝑐} → 𝐶 = {𝑐})
50 dfsn2 4138 . . . . . . . . . . . . . . . . . 18 {𝑐} = {𝑐, 𝑐}
5149, 50syl6eq 2660 . . . . . . . . . . . . . . . . 17 (𝐶 = {𝑐} → 𝐶 = {𝑐, 𝑐})
5251adantl 481 . . . . . . . . . . . . . . . 16 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 𝐶 = {𝑐}) → 𝐶 = {𝑐, 𝑐})
53 preq1 4212 . . . . . . . . . . . . . . . . . 18 (𝑎 = 𝑐 → {𝑎, 𝑏} = {𝑐, 𝑏})
5453eqeq2d 2620 . . . . . . . . . . . . . . . . 17 (𝑎 = 𝑐 → (𝐶 = {𝑎, 𝑏} ↔ 𝐶 = {𝑐, 𝑏}))
55 preq2 4213 . . . . . . . . . . . . . . . . . 18 (𝑏 = 𝑐 → {𝑐, 𝑏} = {𝑐, 𝑐})
5655eqeq2d 2620 . . . . . . . . . . . . . . . . 17 (𝑏 = 𝑐 → (𝐶 = {𝑐, 𝑏} ↔ 𝐶 = {𝑐, 𝑐}))
5754, 56rspc2ev 3295 . . . . . . . . . . . . . . . 16 ((𝑐𝑉𝑐𝑉𝐶 = {𝑐, 𝑐}) → ∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏})
5848, 48, 52, 57syl3anc 1318 . . . . . . . . . . . . . . 15 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 𝐶 = {𝑐}) → ∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏})
59 r2ex 3043 . . . . . . . . . . . . . . 15 (∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏} ↔ ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏}))
6058, 59sylib 207 . . . . . . . . . . . . . 14 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ 𝐶 = {𝑐}) → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏}))
6160ex 449 . . . . . . . . . . . . 13 (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → (𝐶 = {𝑐} → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
6261exlimdv 1848 . . . . . . . . . . . 12 (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → (∃𝑐 𝐶 = {𝑐} → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
6338, 62sylbid 229 . . . . . . . . . . 11 (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → ((#‘𝐶) = 1 → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
6463adantr 480 . . . . . . . . . 10 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → ((#‘𝐶) = 1 → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
6537, 64jaod 394 . . . . . . . . 9 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → (((#‘𝐶) = 0 ∨ (#‘𝐶) = 1) → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
6628, 65syld 46 . . . . . . . 8 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → ((#‘𝐶) < 2 → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
67 hash2sspr 13124 . . . . . . . . . . . . 13 ((𝐶 ∈ 𝒫 𝑉 ∧ (#‘𝐶) = 2) → ∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏})
6867ex 449 . . . . . . . . . . . 12 (𝐶 ∈ 𝒫 𝑉 → ((#‘𝐶) = 2 → ∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏}))
6968adantr 480 . . . . . . . . . . 11 ((𝐶 ∈ 𝒫 𝑉𝐶 ≠ ∅) → ((#‘𝐶) = 2 → ∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏}))
7029, 69sylbi 206 . . . . . . . . . 10 (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → ((#‘𝐶) = 2 → ∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏}))
7170, 59syl6ib 240 . . . . . . . . 9 (𝐶 ∈ (𝒫 𝑉 ∖ {∅}) → ((#‘𝐶) = 2 → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
7271adantr 480 . . . . . . . 8 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → ((#‘𝐶) = 2 → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
7366, 72jaod 394 . . . . . . 7 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → (((#‘𝐶) < 2 ∨ (#‘𝐶) = 2) → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
7425, 73sylbid 229 . . . . . 6 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ∈ ℕ0) → ((#‘𝐶) ≤ 2 → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
7574impancom 455 . . . . 5 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ≤ 2) → ((#‘𝐶) ∈ ℕ0 → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏})))
7620, 75mpd 15 . . . 4 ((𝐶 ∈ (𝒫 𝑉 ∖ {∅}) ∧ (#‘𝐶) ≤ 2) → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏}))
7715, 76sylbi 206 . . 3 (𝐶 ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏}))
7812, 77syl 17 . 2 ((𝐺 ∈ UPGraph ∧ 𝐶𝐸) → ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝐶 = {𝑎, 𝑏}))
7978, 59sylibr 223 1 ((𝐺 ∈ UPGraph ∧ 𝐶𝐸) → ∃𝑎𝑉𝑏𝑉 𝐶 = {𝑎, 𝑏})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127   class class class wbr 4583  dom cdm 5038  ran crn 5039  ⟶wf 5800  ‘cfv 5804  Fincfn 7841  ℝcr 9814  0cc0 9815  1c1 9816   < clt 9953   ≤ cle 9954  2c2 10947  ℕ0cn0 11169  #chash 12979  Vtxcvtx 25673  iEdgciedg 25674   UPGraph cupgr 25747  Edgcedga 25792 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-hash 12980  df-upgr 25749  df-edga 25793 This theorem is referenced by:  upgredg2vtx  25814  upgredgpr  25815
 Copyright terms: Public domain W3C validator