Theorem umgr0e 29142

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 6883	. . 3 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
3		f1f 6805	. . 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 2735	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
7		eqid 2735	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
8	6, 7	isumgr 29127	. . 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 1537   ∈ wcel 2106  {crab 3433   ∖ cdif 3960  ∅c0 4339  𝒫 cpw 4605  {csn 4631  dom cdm 5689  ⟶wf 6559  –1-1→wf1 6560  ‘cfv 6563  2c2 12319  ♯chash 14366  Vtxcvtx 29028  iEdgciedg 29029  UMGraphcumgr 29113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fv 6571  df-umgr 29115

This theorem is referenced by:  upgr0e  29143