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Mirrors > Home > MPE Home > Th. List > uhgr0e | Structured version Visualization version GIF version |
Description: The empty graph, with vertices but no edges, is a hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 25-Nov-2020.) |
Ref | Expression |
---|---|
uhgr0e.g | ⊢ (𝜑 → 𝐺 ∈ 𝑊) |
uhgr0e.e | ⊢ (𝜑 → (iEdg‘𝐺) = ∅) |
Ref | Expression |
---|---|
uhgr0e | ⊢ (𝜑 → 𝐺 ∈ UHGraph) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f0 6762 | . . 3 ⊢ ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) | |
2 | dm0 5915 | . . . 4 ⊢ dom ∅ = ∅ | |
3 | 2 | feq2i 6699 | . . 3 ⊢ (∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
4 | 1, 3 | mpbir 230 | . 2 ⊢ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) |
5 | uhgr0e.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
6 | eqid 2733 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
7 | eqid 2733 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
8 | 6, 7 | isuhgr 28287 | . . . 4 ⊢ (𝐺 ∈ 𝑊 → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
9 | 5, 8 | syl 17 | . . 3 ⊢ (𝜑 → (𝐺 ∈ UHGraph ↔ (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
10 | uhgr0e.e | . . . 4 ⊢ (𝜑 → (iEdg‘𝐺) = ∅) | |
11 | id 22 | . . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → (iEdg‘𝐺) = ∅) | |
12 | dmeq 5898 | . . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅) | |
13 | 11, 12 | feq12d 6695 | . . . 4 ⊢ ((iEdg‘𝐺) = ∅ → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
14 | 10, 13 | syl 17 | . . 3 ⊢ (𝜑 → ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
15 | 9, 14 | bitrd 279 | . 2 ⊢ (𝜑 → (𝐺 ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
16 | 4, 15 | mpbiri 258 | 1 ⊢ (𝜑 → 𝐺 ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ∖ cdif 3943 ∅c0 4320 𝒫 cpw 4598 {csn 4624 dom cdm 5672 ⟶wf 6531 ‘cfv 6535 Vtxcvtx 28223 iEdgciedg 28224 UHGraphcuhgr 28283 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5295 ax-nul 5302 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-opab 5207 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-fv 6543 df-uhgr 28285 |
This theorem is referenced by: uhgr0vb 28299 |
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