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Theorem upgrle2 28365
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 484 . 2 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UPGraph)
2 upgruhgr 28362 . . . . 5 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
3 upgrle2.i . . . . . 6 𝐼 = (iEdg‘𝐺)
43uhgrfun 28326 . . . . 5 (𝐺 ∈ UHGraph → Fun 𝐼)
52, 4syl 17 . . . 4 (𝐺 ∈ UPGraph → Fun 𝐼)
65funfnd 6580 . . 3 (𝐺 ∈ UPGraph → 𝐼 Fn dom 𝐼)
76adantr 482 . 2 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼)
8 simpr 486 . 2 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom 𝐼)
9 eqid 2733 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
109, 3upgrle 28350 . 2 ((𝐺 ∈ UPGraph ∧ 𝐼 Fn dom 𝐼𝑋 ∈ dom 𝐼) → (♯‘(𝐼𝑋)) ≤ 2)
111, 7, 8, 10syl3anc 1372 1 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼𝑋)) ≤ 2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107   class class class wbr 5149  dom cdm 5677  Fun wfun 6538   Fn wfn 6539  cfv 6544  cle 11249  2c2 12267  chash 14290  Vtxcvtx 28256  iEdgciedg 28257  UHGraphcuhgr 28316  UPGraphcupgr 28340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-uhgr 28318  df-upgr 28342
This theorem is referenced by:  upgr2pthnlp  28989
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