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Theorem isuslgra 25872
 Description: The property of being an undirected simple graph with loops. (Contributed by Alexander van der Vekens, 10-Aug-2017.)
Assertion
Ref Expression
isuslgra ((𝑉𝑊𝐸𝑋) → (𝑉 USLGrph 𝐸𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
Distinct variable groups:   𝑥,𝐸   𝑥,𝑉
Allowed substitution hints:   𝑊(𝑥)   𝑋(𝑥)

Proof of Theorem isuslgra
Dummy variables 𝑣 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1eq1 6009 . . . 4 (𝑒 = 𝐸 → (𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ 𝐸:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
21adantl 481 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ 𝐸:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
3 dmeq 5246 . . . . 5 (𝑒 = 𝐸 → dom 𝑒 = dom 𝐸)
43adantl 481 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → dom 𝑒 = dom 𝐸)
5 f1eq2 6010 . . . 4 (dom 𝑒 = dom 𝐸 → (𝐸:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
64, 5syl 17 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝐸:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
7 simpl 472 . . . . . 6 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝑣 = 𝑉)
87pweqd 4113 . . . . 5 ((𝑣 = 𝑉𝑒 = 𝐸) → 𝒫 𝑣 = 𝒫 𝑉)
98difeq1d 3689 . . . 4 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝒫 𝑣 ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
10 rabeq 3166 . . . 4 ((𝒫 𝑣 ∖ {∅}) = (𝒫 𝑉 ∖ {∅}) → {𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
11 f1eq3 6011 . . . 4 ({𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} = {𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
129, 10, 113syl 18 . . 3 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
132, 6, 123bitrd 293 . 2 ((𝑣 = 𝑉𝑒 = 𝐸) → (𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} ↔ 𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
14 df-uslgra 25861 . 2 USLGrph = {⟨𝑣, 𝑒⟩ ∣ 𝑒:dom 𝑒1-1→{𝑥 ∈ (𝒫 𝑣 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}}
1513, 14brabga 4914 1 ((𝑉𝑊𝐸𝑋) → (𝑉 USLGrph 𝐸𝐸:dom 𝐸1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {crab 2900   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  –1-1→wf1 5801  ‘cfv 5804   ≤ cle 9954  2c2 10947  #chash 12979   USLGrph cuslg 25858 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-uslgra 25861 This theorem is referenced by:  uslgraf  25874  uslisushgra  25892  uslisumgra  25893  usisuslgra  25894  uslgra1  25901
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