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Theorem upgrle2 28632
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 481 . 2 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UPGraph)
2 upgruhgr 28629 . . . . 5 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
3 upgrle2.i . . . . . 6 𝐼 = (iEdg‘𝐺)
43uhgrfun 28593 . . . . 5 (𝐺 ∈ UHGraph → Fun 𝐼)
52, 4syl 17 . . . 4 (𝐺 ∈ UPGraph → Fun 𝐼)
65funfnd 6578 . . 3 (𝐺 ∈ UPGraph → 𝐼 Fn dom 𝐼)
76adantr 479 . 2 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼)
8 simpr 483 . 2 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom 𝐼)
9 eqid 2730 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
109, 3upgrle 28617 . 2 ((𝐺 ∈ UPGraph ∧ 𝐼 Fn dom 𝐼𝑋 ∈ dom 𝐼) → (♯‘(𝐼𝑋)) ≤ 2)
111, 7, 8, 10syl3anc 1369 1 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼𝑋)) ≤ 2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104   class class class wbr 5147  dom cdm 5675  Fun wfun 6536   Fn wfn 6537  cfv 6542  cle 11253  2c2 12271  chash 14294  Vtxcvtx 28523  iEdgciedg 28524  UHGraphcuhgr 28583  UPGraphcupgr 28607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-fv 6550  df-uhgr 28585  df-upgr 28609
This theorem is referenced by:  upgr2pthnlp  29256
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