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| Mirrors > Home > MPE Home > Th. List > umgr0e | Structured version Visualization version GIF version | ||
| 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.) | 
| Ref | Expression | 
|---|---|
| umgr0e.g | ⊢ (𝜑 → 𝐺 ∈ 𝑊) | 
| umgr0e.e | ⊢ (𝜑 → (iEdg‘𝐺) = ∅) | 
| Ref | Expression | 
|---|---|
| umgr0e | ⊢ (𝜑 → 𝐺 ∈ UMGraph) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | umgr0e.e | . . . 4 ⊢ (𝜑 → (iEdg‘𝐺) = ∅) | |
| 2 | 1 | f10d 6881 | . . 3 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}) | 
| 3 | f1f 6803 | . . 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 29113 | . . 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) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 {crab 3435 ∖ cdif 3947 ∅c0 4332 𝒫 cpw 4599 {csn 4625 dom cdm 5684 ⟶wf 6556 –1-1→wf1 6557 ‘cfv 6560 2c2 12322 ♯chash 14370 Vtxcvtx 29014 iEdgciedg 29015 UMGraphcumgr 29099 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fv 6568 df-umgr 29101 | 
| This theorem is referenced by: upgr0e 29129 | 
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