Theorem upgrle2 27378

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 27375	. . . . 5 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
3		upgrle2.i	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
4	3	uhgrfun 27339	. . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
5	2, 4	syl 17	. . . 4 ⊢ (𝐺 ∈ UPGraph → Fun 𝐼)
6	5	funfnd 6449	. . 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 2738	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
10	9, 3	upgrle 27363	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐼 Fn dom 𝐼 ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) ≤ 2)
11	1, 7, 8, 10	syl3anc 1369	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) ≤ 2)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108   class class class wbr 5070  dom cdm 5580  Fun wfun 6412   Fn wfn 6413  ‘cfv 6418   ≤ cle 10941  2c2 11958  ♯chash 13972  Vtxcvtx 27269  iEdgciedg 27270  UHGraphcuhgr 27329  UPGraphcupgr 27353

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-uhgr 27331  df-upgr 27355

This theorem is referenced by:  upgr2pthnlp  28001