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Theorem upgrle 26317
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 26314 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
43ffvelrnda 6583 . . 3 (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴) ∧ 𝐹𝐴) → (𝐸𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
543impa 1137 . 2 ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴𝐹𝐴) → (𝐸𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
6 fveq2 6409 . . . . 5 (𝑥 = (𝐸𝐹) → (♯‘𝑥) = (♯‘(𝐸𝐹)))
76breq1d 4851 . . . 4 (𝑥 = (𝐸𝐹) → ((♯‘𝑥) ≤ 2 ↔ (♯‘(𝐸𝐹)) ≤ 2))
87elrab 3554 . . 3 ((𝐸𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ ((𝐸𝐹) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸𝐹)) ≤ 2))
98simprbi 491 . 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 385  w3a 1108   = wceq 1653  wcel 2157  {crab 3091  cdif 3764  c0 4113  𝒫 cpw 4347  {csn 4366   class class class wbr 4841   Fn wfn 6094  cfv 6099  cle 10362  2c2 11364  chash 13366  Vtxcvtx 26223  iEdgciedg 26224  UPGraphcupgr 26307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pr 5095
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-br 4842  df-opab 4904  df-id 5218  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-fv 6107  df-upgr 26309
This theorem is referenced by:  upgrfi  26318  upgrex  26319  upgrle2  26332  subupgr  26513  upgrewlkle2  26848
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