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Theorem upgrle2 26905
 Description: An edge of an undirected pseudograph has at most two ends. (Contributed by AV, 6-Feb-2021.)
Hypothesis
Ref Expression
upgrle2.i 𝐼 = (iEdg‘𝐺)
Assertion
Ref Expression
upgrle2 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼𝑋)) ≤ 2)

Proof of Theorem upgrle2
StepHypRef Expression
1 simpl 486 . 2 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UPGraph)
2 upgruhgr 26902 . . . . 5 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
3 upgrle2.i . . . . . 6 𝐼 = (iEdg‘𝐺)
43uhgrfun 26866 . . . . 5 (𝐺 ∈ UHGraph → Fun 𝐼)
52, 4syl 17 . . . 4 (𝐺 ∈ UPGraph → Fun 𝐼)
65funfnd 6355 . . 3 (𝐺 ∈ UPGraph → 𝐼 Fn dom 𝐼)
76adantr 484 . 2 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼)
8 simpr 488 . 2 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom 𝐼)
9 eqid 2798 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
109, 3upgrle 26890 . 2 ((𝐺 ∈ UPGraph ∧ 𝐼 Fn dom 𝐼𝑋 ∈ dom 𝐼) → (♯‘(𝐼𝑋)) ≤ 2)
111, 7, 8, 10syl3anc 1368 1 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼𝑋)) ≤ 2)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   class class class wbr 5030  dom cdm 5519  Fun wfun 6318   Fn wfn 6319  ‘cfv 6324   ≤ cle 10667  2c2 11682  ♯chash 13688  Vtxcvtx 26796  iEdgciedg 26797  UHGraphcuhgr 26856  UPGraphcupgr 26880 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-uhgr 26858  df-upgr 26882 This theorem is referenced by:  upgr2pthnlp  27528
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