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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 6474 | . . 3 ⊢ (𝜑 → (iEdg‘𝐺):dom (iEdg‘𝐺)–1-1→{𝑥 ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) ∣ (♯‘𝑥) = 2}) |
3 | f1f 6401 | . . 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 2771 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
7 | eqid 2771 | . . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
8 | 6, 7 | isumgr 26598 | . . 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 249 | 1 ⊢ (𝜑 → 𝐺 ∈ UMGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1508 ∈ wcel 2051 {crab 3085 ∖ cdif 3819 ∅c0 4172 𝒫 cpw 4416 {csn 4435 dom cdm 5403 ⟶wf 6181 –1-1→wf1 6182 ‘cfv 6185 2c2 11493 ♯chash 13503 Vtxcvtx 26499 iEdgciedg 26500 UMGraphcumgr 26584 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-sep 5056 ax-nul 5063 ax-pr 5182 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ral 3086 df-rex 3087 df-rab 3090 df-v 3410 df-sbc 3675 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-opab 4988 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fv 6193 df-umgr 26586 |
This theorem is referenced by: upgr0e 26614 |
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