Theorem uslisumgra 25893

Description: An undirected simple graph with loops is an undirected multigraph. (Contributed by Alexander van der Vekens, 10-Aug-2017.)

Assertion

Ref	Expression
uslisumgra	⊢ (𝑉 USLGrph 𝐸 → 𝑉 UMGrph 𝐸)

Proof of Theorem uslisumgra

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uslgrav 25866	. 2 ⊢ (𝑉 USLGrph 𝐸 → (𝑉 ∈ V ∧ 𝐸 ∈ V))
2		isuslgra 25872	. . . 4 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 USLGrph 𝐸 ↔ 𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
3		f1f 6014	. . . 4 ⊢ (𝐸:dom 𝐸–1-1→{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2} → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2})
4	2, 3	syl6bi 242	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 USLGrph 𝐸 → 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
5		isumgra 25844	. . 3 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 UMGrph 𝐸 ↔ 𝐸:dom 𝐸⟶{𝑥 ∈ (𝒫 𝑉 ∖ {∅}) ∣ (#‘𝑥) ≤ 2}))
6	4, 5	sylibrd 248	. 2 ⊢ ((𝑉 ∈ V ∧ 𝐸 ∈ V) → (𝑉 USLGrph 𝐸 → 𝑉 UMGrph 𝐸))
7	1, 6	mpcom 37	1 ⊢ (𝑉 USLGrph 𝐸 → 𝑉 UMGrph 𝐸)

Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  {crab 2900  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  𝒫 cpw 4108  {csn 4125   class class class wbr 4583  dom cdm 5038  ⟶wf 5800  –1-1→wf1 5801  ‘cfv 5804   ≤ cle 9954  2c2 10947  #chash 12979   UMGrph cumg 25841   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-ral 2901  df-rex 2902  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-xp 5044  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-umgra 25842  df-uslgra 25861

This theorem is referenced by:  usisumgra  25895  uslgraun  25903