Theorem umgr0e 29090

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 6802	. . 3 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2})
3		f1f 6724	. . 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 2733	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
7		eqid 2733	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
8	6, 7	isumgr 29075	. . 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 3396   ∖ cdif 3895  ∅c0 4282  𝒫 cpw 4549  {csn 4575  dom cdm 5619  ⟶wf 6482  –1-1→wf1 6483  ‘cfv 6486  2c2 12187  ♯chash 14239  Vtxcvtx 28976  iEdgciedg 28977  UMGraphcumgr 29061

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 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

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 2537  df-clab 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fv 6494  df-umgr 29063

This theorem is referenced by:  upgr0e  29091