Theorem upgrle 29175

Description: An edge of an undirected pseudograph has at most two ends. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)

Hypotheses

Ref	Expression
isupgr.v	⊢ 𝑉 = (Vtx‘𝐺)
isupgr.e	⊢ 𝐸 = (iEdg‘𝐺)

Assertion

Ref	Expression
upgrle	⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (♯‘(𝐸‘𝐹)) ≤ 2)

Proof of Theorem upgrle

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isupgr.v	. . . . 5 ⊢ 𝑉 = (Vtx‘𝐺)
2		isupgr.e	. . . . 5 ⊢ 𝐸 = (iEdg‘𝐺)
3	1, 2	upgrfn 29172	. . . 4 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴) → 𝐸:𝐴⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
4	3	ffvelcdmda 7038	. . 3 ⊢ (((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴) ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
5	4	3impa 1110	. 2 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (𝐸‘𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2})
6		fveq2 6842	. . . . 5 ⊢ (𝑥 = (𝐸‘𝐹) → (♯‘𝑥) = (♯‘(𝐸‘𝐹)))
7	6	breq1d 5110	. . . 4 ⊢ (𝑥 = (𝐸‘𝐹) → ((♯‘𝑥) ≤ 2 ↔ (♯‘(𝐸‘𝐹)) ≤ 2))
8	7	elrab 3648	. . 3 ⊢ ((𝐸‘𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} ↔ ((𝐸‘𝐹) ∈ (𝒫 𝑉 ∖ {∅}) ∧ (♯‘(𝐸‘𝐹)) ≤ 2))
9	8	simprbi 497	. 2 ⊢ ((𝐸‘𝐹) ∈ {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (♯‘𝑥) ≤ 2} → (♯‘(𝐸‘𝐹)) ≤ 2)
10	5, 9	syl 17	1 ⊢ ((𝐺 ∈ UPGraph ∧ 𝐸 Fn 𝐴 ∧ 𝐹 ∈ 𝐴) → (♯‘(𝐸‘𝐹)) ≤ 2)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  {crab 3401   ∖ cdif 3900  ∅c0 4287  𝒫 cpw 4556  {csn 4582   class class class wbr 5100   Fn wfn 6495  ‘cfv 6500   ≤ cle 11179  2c2 12212  ♯chash 14265  Vtxcvtx 29081  iEdgciedg 29082  UPGraphcupgr 29165

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-upgr 29167

This theorem is referenced by:  upgrfi  29176  upgrex  29177  upgrle2  29190  subupgr  29372  upgrewlkle2  29692