Theorem upgrle2 29398

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 487	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UPGraph)
2		upgruhgr 29395	. . . . 5 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
3		upgrle2.i	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
4	3	uhgrfun 29359	. . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
5	2, 4	syl 18	. . . 4 ⊢ (𝐺 ∈ UPGraph → Fun 𝐼)
6	5	funfnd 6570	. . 3 ⊢ (𝐺 ∈ UPGraph → 𝐼 Fn dom 𝐼)
7	6	adantr 485	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼)
8		simpr 489	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom 𝐼)
9		eqid 2769	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
10	9, 3	upgrle 29383	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐼 Fn dom 𝐼 ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) ≤ 2)
11	1, 7, 8, 10	syl3anc 1396	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) ≤ 2)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149   class class class wbr 5113  dom cdm 5664  Fun wfun 6533   Fn wfn 6534  ‘cfv 6539   ≤ cle 11246  2c2 12297  ♯chash 14368  Vtxcvtx 29289  iEdgciedg 29290  UHGraphcuhgr 29349  UPGraphcupgr 29373

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5273  ax-pr 5407

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-id 5559  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-fv 6547  df-uhgr 29351  df-upgr 29375

This theorem is referenced by:  upgr2pthnlp  30024