Theorem upgrle2 29594

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 488	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐺 ∈ UPGraph)
2		upgruhgr 29591	. . . . 5 ⊢ (𝐺 ∈ UPGraph → 𝐺 ∈ UHGraph)
3		upgrle2.i	. . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺)
4	3	uhgrfun 29555	. . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐼)
5	2, 4	syl 18	. . . 4 ⊢ (𝐺 ∈ UPGraph → Fun 𝐼)
6	5	funfnd 6567	. . 3 ⊢ (𝐺 ∈ UPGraph → 𝐼 Fn dom 𝐼)
7	6	adantr 486	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝐼 Fn dom 𝐼)
8		simpr 490	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → 𝑋 ∈ dom 𝐼)
9		eqid 2760	. . 3 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
10	9, 3	upgrle 29579	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐼 Fn dom 𝐼 ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) ≤ 2)
11	1, 7, 8, 10	syl3anc 1398	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝑋 ∈ dom 𝐼) → (♯‘(𝐼‘𝑋)) ≤ 2)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   class class class wbr 5103  dom cdm 5655  Fun wfun 6529   Fn wfn 6530  ‘cfv 6535   ≤ cle 11293  2c2 12344  ♯chash 14419  Vtxcvtx 29485  iEdgciedg 29486  UHGraphcuhgr 29545  UPGraphcupgr 29569

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6491  df-fun 6537  df-fn 6538  df-f 6539  df-fv 6543  df-uhgr 29547  df-upgr 29571

This theorem is used by:  upgr2pthnlp  30229