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Theorem upgrle 26791
Description: An edge of an undirected pseudograph has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)
Hypotheses
Ref Expression
isupgr.v 𝑉 = (Vtx‘𝐺)
isupgr.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
upgrle ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (♯‘(𝐸𝐹)) ≤ 2)

Proof of Theorem upgrle
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isupgr.v . . . . 5 𝑉 = (Vtx‘𝐺)
2 isupgr.e . . . . 5 𝐸 = (iEdg‘𝐺)
31, 2upgrfn 26788 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
43ffvelrnda 6846 . . 3 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴) ∧ 𝐹𝐴) → (𝐸𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
543impa 1104 . 2 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
6 fveq2 6666 . . . . 5 (𝑥 = (𝐸𝐹) → (♯‘𝑥) = (♯‘(𝐸𝐹)))
76breq1d 5072 . . . 4 (𝑥 = (𝐸𝐹) → ((♯‘𝑥) ≤ 2 ↔ (♯‘(𝐸𝐹)) ≤ 2))
87elrab 3683 . . 3 ((𝐸𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ ((𝐸𝐹) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝐹)) ≤ 2))
98simprbi 497 . 2 ((𝐸𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (♯‘(𝐸𝐹)) ≤ 2)
105, 9syl 17 1 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (♯‘(𝐸𝐹)) ≤ 2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081   = wceq 1530  wcel 2107  {crab 3146  cdif 3936  c0 4294  𝒫 cpw 4541  {csn 4563   class class class wbr 5062   Fn wfn 6346  cfv 6351  cle 10668  2c2 11684  chash 13683  Vtxcvtx 26697  iEdgciedg 26698  UPGraphcupgr 26781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-upgr 26783
This theorem is referenced by:  upgrfi  26792  upgrex  26793  upgrle2  26806  subupgr  26985  upgrewlkle2  27304
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