Theorem umgr0e 29441

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 6857	. . 3 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
3		f1f 6776	. . 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 18	. 2 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
5		umgr0e.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝑊)
6		eqid 2763	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
7		eqid 2763	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
8	6, 7	isumgr 29426	. . 3 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ UMGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
9	5, 8	syl 18	. 2 ⊢ (𝜑 → (𝐺 ∈ UMGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}))
10	4, 9	mpbird 260	1 ⊢ (𝜑 → 𝐺 ∈ UMGraph)

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2143  {crab 3416   ∖ cdif 3903  ∅c0 4287  𝒫 cpw 4563  {csn 4590  dom cdm 5663  ⟶wf 6534  –1-1→wf1 6535  ‘cfv 6538  2c2 12296  ♯chash 14368  Vtxcvtx 29327  iEdgciedg 29328  UMGraphcumgr 29412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fv 6546  df-umgr 29414

This theorem is referenced by:  upgr0e  29442