Theorem upgrle2 29127

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) → (♯‘(𝐼‘𝑋)) ≤ 2)

Proof of Theorem upgrle2

Step	Hyp	Ref	Expression
1		simpl 482	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UPGraph)
2		upgruhgr 29124	. . . . 5 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
3		upgrle2.i	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
4	3	uhgrfun 29088	. . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
5	2, 4	syl 17	. . . 4 ⊢ (𝐺 ∈ UPGraph → Fun 𝐼)
6	5	funfnd 6521	. . 3 ⊢ (𝐺 ∈ UPGraph → 𝐼 Fn dom 𝐼)
7	6	adantr 480	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼)
8		simpr 484	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom 𝐼)
9		eqid 2734	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
10	9, 3	upgrle 29112	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐼 Fn dom 𝐼 ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) ≤ 2)
11	1, 7, 8, 10	syl3anc 1373	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) ≤ 2)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   class class class wbr 5096  dom cdm 5622  Fun wfun 6484   Fn wfn 6485  ‘cfv 6490   ≤ cle 11165  2c2 12198  ♯chash 14251  Vtxcvtx 29018  iEdgciedg 29019  UHGraphcuhgr 29078  UPGraphcupgr 29102

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-uhgr 29080  df-upgr 29104

This theorem is referenced by:  upgr2pthnlp  29754