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Theorem upgrle2 26817
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 483 . 2 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UPGraph)
2 upgruhgr 26814 . . . . 5 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
3 upgrle2.i . . . . . 6 𝐼 = (iEdg‘𝐺)
43uhgrfun 26778 . . . . 5 (𝐺 ∈ UHGraph → Fun 𝐼)
52, 4syl 17 . . . 4 (𝐺 ∈ UPGraph → Fun 𝐼)
65funfnd 6379 . . 3 (𝐺 ∈ UPGraph → 𝐼 Fn dom 𝐼)
76adantr 481 . 2 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼)
8 simpr 485 . 2 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom 𝐼)
9 eqid 2818 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
109, 3upgrle 26802 . 2 ((𝐺 ∈ UPGraph ∧ 𝐼 Fn dom 𝐼𝑋 ∈ dom 𝐼) → (♯‘(𝐼𝑋)) ≤ 2)
111, 7, 8, 10syl3anc 1363 1 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼𝑋)) ≤ 2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105   class class class wbr 5057  dom cdm 5548  Fun wfun 6342   Fn wfn 6343  cfv 6348  cle 10664  2c2 11680  chash 13678  Vtxcvtx 26708  iEdgciedg 26709  UHGraphcuhgr 26768  UPGraphcupgr 26792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-uhgr 26770  df-upgr 26794
This theorem is referenced by:  upgr2pthnlp  27440
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