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Mirrors > Home > MPE Home > Th. List > uhgrf | Structured version Visualization version GIF version |
Description: The edge function of an undirected hypergraph is a function into the power set of the set of vertices. (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 9-Oct-2020.) |
Ref | Expression |
---|---|
uhgrf.v | ⊢ 𝑉 = (Vtx‘𝐺) |
uhgrf.e | ⊢ 𝐸 = (iEdg‘𝐺) |
Ref | Expression |
---|---|
uhgrf | ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgrf.v | . . 3 ⊢ 𝑉 = (Vtx‘𝐺) | |
2 | uhgrf.e | . . 3 ⊢ 𝐸 = (iEdg‘𝐺) | |
3 | 1, 2 | isuhgr 26831 | . 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
4 | 3 | ibi 269 | 1 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∖ cdif 3921 ∅c0 4279 𝒫 cpw 4525 {csn 4553 dom cdm 5541 ⟶wf 6337 ‘cfv 6341 Vtxcvtx 26767 iEdgciedg 26768 UHGraphcuhgr 26827 |
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 2793 ax-nul 5196 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3488 df-sbc 3764 df-dif 3927 df-un 3929 df-in 3931 df-ss 3940 df-nul 4280 df-if 4454 df-pw 4527 df-sn 4554 df-pr 4556 df-op 4560 df-uni 4825 df-br 5053 df-opab 5115 df-rel 5548 df-cnv 5549 df-co 5550 df-dm 5551 df-rn 5552 df-iota 6300 df-fun 6343 df-fn 6344 df-f 6345 df-fv 6349 df-uhgr 26829 |
This theorem is referenced by: uhgrss 26835 uhgrfun 26837 uhgrn0 26838 uhgr0vb 26843 uhgrun 26845 uhgredgn0 26899 2pthon3v 27707 |
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