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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 6641 | . . 3 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
3 | f1f 6568 | . . 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 2818 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
7 | eqid 2818 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
8 | 6, 7 | isumgr 26807 | . . 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 258 | 1 ⊢ (𝜑 → 𝐺 ∈ UMGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1528 ∈ wcel 2105 {crab 3139 ∖ cdif 3930 ∅c0 4288 𝒫 cpw 4535 {csn 4557 dom cdm 5548 ⟶wf 6344 –1-1→wf1 6345 ‘cfv 6348 2c2 11680 ♯chash 13678 Vtxcvtx 26708 iEdgciedg 26709 UMGraphcumgr 26793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fv 6356 df-umgr 26795 |
This theorem is referenced by: upgr0e 26823 |
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