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Theorem edgupgr 26930
 Description: Properties of an edge of a pseudograph. (Contributed by AV, 8-Nov-2020.)
Assertion
Ref Expression
edgupgr ((𝐺 ∈ UPGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (♯‘𝐸) ≤ 2))

Proof of Theorem edgupgr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 edgval 26845 . . . . 5 (Edg‘𝐺) = ran (iEdg‘𝐺)
21a1i 11 . . . 4 (𝐺 ∈ UPGraph → (Edg‘𝐺) = ran (iEdg‘𝐺))
32eleq2d 2878 . . 3 (𝐺 ∈ UPGraph → (𝐸 ∈ (Edg‘𝐺) ↔ 𝐸 ∈ ran (iEdg‘𝐺)))
4 eqid 2801 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
5 eqid 2801 . . . . . . 7 (iEdg‘𝐺) = (iEdg‘𝐺)
64, 5upgrf 26882 . . . . . 6 (𝐺 ∈ UPGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
76frnd 6498 . . . . 5 (𝐺 ∈ UPGraph → ran (iEdg‘𝐺) ⊆ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
87sseld 3917 . . . 4 (𝐺 ∈ UPGraph → (𝐸 ∈ ran (iEdg‘𝐺) → 𝐸 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2}))
9 fveq2 6649 . . . . . . 7 (𝑥 = 𝐸 → (♯‘𝑥) = (♯‘𝐸))
109breq1d 5043 . . . . . 6 (𝑥 = 𝐸 → ((♯‘𝑥) ≤ 2 ↔ (♯‘𝐸) ≤ 2))
1110elrab 3631 . . . . 5 (𝐸 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ (𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝐸) ≤ 2))
12 eldifsn 4683 . . . . . . . . 9 (𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅))
1312biimpi 219 . . . . . . . 8 (𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅))
1413anim1i 617 . . . . . . 7 ((𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝐸) ≤ 2) → ((𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅) ∧ (♯‘𝐸) ≤ 2))
15 df-3an 1086 . . . . . . 7 ((𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (♯‘𝐸) ≤ 2) ↔ ((𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅) ∧ (♯‘𝐸) ≤ 2))
1614, 15sylibr 237 . . . . . 6 ((𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝐸) ≤ 2) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (♯‘𝐸) ≤ 2))
1716a1i 11 . . . . 5 (𝐺 ∈ UPGraph → ((𝐸 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ (♯‘𝐸) ≤ 2) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (♯‘𝐸) ≤ 2)))
1811, 17syl5bi 245 . . . 4 (𝐺 ∈ UPGraph → (𝐸 ∈ {𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (♯‘𝐸) ≤ 2)))
198, 18syld 47 . . 3 (𝐺 ∈ UPGraph → (𝐸 ∈ ran (iEdg‘𝐺) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (♯‘𝐸) ≤ 2)))
203, 19sylbid 243 . 2 (𝐺 ∈ UPGraph → (𝐸 ∈ (Edg‘𝐺) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (♯‘𝐸) ≤ 2)))
2120imp 410 1 ((𝐺 ∈ UPGraph ∧ 𝐸 ∈ (Edg‘𝐺)) → (𝐸 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝐸 ≠ ∅ ∧ (♯‘𝐸) ≤ 2))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112   ≠ wne 2990  {crab 3113   ∖ cdif 3881  ∅c0 4246  𝒫 cpw 4500  {csn 4528   class class class wbr 5033  dom cdm 5523  ran crn 5524  ‘cfv 6328   ≤ cle 10669  2c2 11684  ♯chash 13690  Vtxcvtx 26792  iEdgciedg 26793  Edgcedg 26843  UPGraphcupgr 26876 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 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 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 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-edg 26844  df-upgr 26878 This theorem is referenced by:  upgrres1  27106
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