Theorem upgrle2 29140

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 29137	. . . . 5 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
3		upgrle2.i	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
4	3	uhgrfun 29101	. . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
5	2, 4	syl 17	. . . 4 ⊢ (𝐺 ∈ UPGraph → Fun 𝐼)
6	5	funfnd 6609	. . 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 2740	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
10	9, 3	upgrle 29125	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐼 Fn dom 𝐼 ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) ≤ 2)
11	1, 7, 8, 10	syl3anc 1371	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) ≤ 2)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108   class class class wbr 5166  dom cdm 5700  Fun wfun 6567   Fn wfn 6568  ‘cfv 6573   ≤ cle 11325  2c2 12348  ♯chash 14379  Vtxcvtx 29031  iEdgciedg 29032  UHGraphcuhgr 29091  UPGraphcupgr 29115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-uhgr 29093  df-upgr 29117

This theorem is referenced by:  upgr2pthnlp  29768