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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 6562 | . . 3 ⊢ ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) | |
2 | dm0 5792 | . . . 4 ⊢ dom ∅ = ∅ | |
3 | 2 | feq2i 6508 | . . 3 ⊢ (∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ↔ ∅:∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅})) |
4 | 1, 3 | mpbir 233 | . 2 ⊢ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) |
5 | uhgr0e.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑊) | |
6 | eqid 2823 | . . . . 5 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
7 | eqid 2823 | . . . . 5 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺) | |
8 | 6, 7 | isuhgr 26847 | . . . 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 5774 | . . . . 5 ⊢ ((iEdg‘𝐺) = ∅ → dom (iEdg‘𝐺) = dom ∅) | |
13 | 11, 12 | feq12d 6504 | . . . 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 281 | . 2 ⊢ (𝜑 → (𝐺 ∈ UHGraph ↔ ∅:dom ∅⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))) |
16 | 4, 15 | mpbiri 260 | 1 ⊢ (𝜑 → 𝐺 ∈ UHGraph) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 = wceq 1537 ∈ wcel 2114 ∖ cdif 3935 ∅c0 4293 𝒫 cpw 4541 {csn 4569 dom cdm 5557 ⟶wf 6353 ‘cfv 6357 Vtxcvtx 26783 iEdgciedg 26784 UHGraphcuhgr 26843 |
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 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-fv 6365 df-uhgr 26845 |
This theorem is referenced by: uhgr0vb 26859 |
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