Theorem umgr0e 29183

Description: The empty graph, with vertices but no edges, is a multigraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.)

Hypotheses

Ref	Expression
umgr0e.g	⊢ (𝜑 → 𝐺 ∈ 𝑊)
umgr0e.e	⊢ (𝜑 → (iEdg‘𝐺) = ∅)

Assertion

Ref	Expression
umgr0e	⊢ (𝜑 → 𝐺 ∈ UMGraph)

Proof of Theorem umgr0e

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		umgr0e.e	. . . 4 ⊢ (𝜑 → (iEdg‘𝐺) = ∅)
2	1	f10d 6808	. . 3 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
3		f1f 6730	. . 3 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2} → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
4	2, 3	syl 17	. 2 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
5		umgr0e.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
6		eqid 2736	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
7		eqid 2736	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
8	6, 7	isumgr 29168	. . 3 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ UMGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
9	5, 8	syl 17	. 2 ⊢ (𝜑 → (𝐺 ∈ UMGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
10	4, 9	mpbird 257	1 ⊢ (𝜑 → 𝐺 ∈ UMGraph)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113  {crab 3399   ∖ cdif 3898  ∅c0 4285  𝒫 cpw 4554  {csn 4580  dom cdm 5624  ⟶wf 6488  –1-1→wf1 6489  ‘cfv 6492  2c2 12200  ♯chash 14253  Vtxcvtx 29069  iEdgciedg 29070  UMGraphcumgr 29154

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fv 6500  df-umgr 29156

This theorem is referenced by:  upgr0e  29184