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Theorem upgrle2 25902
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 473 . 2 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UPGraph )
2 upgruhgr 25899 . . . . 5 (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph )
3 upgrle2.i . . . . . 6 𝐼 = (iEdg‘𝐺)
43uhgrfun 25864 . . . . 5 (𝐺 ∈ UHGraph → Fun 𝐼)
52, 4syl 17 . . . 4 (𝐺 ∈ UPGraph → Fun 𝐼)
6 funfn 5879 . . . 4 (Fun 𝐼𝐼 Fn dom 𝐼)
75, 6sylib 208 . . 3 (𝐺 ∈ UPGraph → 𝐼 Fn dom 𝐼)
87adantr 481 . 2 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼)
9 simpr 477 . 2 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom 𝐼)
10 eqid 2621 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
1110, 3upgrle 25888 . 2 ((𝐺 ∈ UPGraph ∧ 𝐼 Fn dom 𝐼𝑋 ∈ dom 𝐼) → (#‘(𝐼𝑋)) ≤ 2)
121, 8, 9, 11syl3anc 1323 1 ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (#‘(𝐼𝑋)) ≤ 2)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987   class class class wbr 4615  dom cdm 5076  Fun wfun 5843   Fn wfn 5844  cfv 5849  cle 10022  2c2 11017  #chash 13060  Vtxcvtx 25781  iEdgciedg 25782   UHGraph cuhgr 25854   UPGraph cupgr 25878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-fv 5857  df-uhgr 25856  df-upgr 25880
This theorem is referenced by:  upgr2pthnlp  26504
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