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| 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 29078 | . 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) | 
| 4 | 3 | ibi 267 | 1 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 ∖ cdif 3947 ∅c0 4332 𝒫 cpw 4599 {csn 4625 dom cdm 5684 ⟶wf 6556 ‘cfv 6560 Vtxcvtx 29014 iEdgciedg 29015 UHGraphcuhgr 29074 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-nul 5305 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-rab 3436 df-v 3481 df-sbc 3788 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-fv 6568 df-uhgr 29076 | 
| This theorem is referenced by: uhgrss 29082 uhgrfun 29084 uhgrn0 29085 uhgr0vb 29090 uhgrun 29092 uhgredgn0 29146 2pthon3v 29964 isubgrvtxuhgr 47855 isubgredg 47857 isubgruhgr 47859 isubgr0uhgr 47864 uhgrimisgrgric 47904 | 
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