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Mirrors > Home > MPE Home > Th. List > isuhgra | Structured version Visualization version GIF version |
Description: The property of being an undirected hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.) |
Ref | Expression |
---|---|
isuhgra | ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝑉 UHGrph 𝐸 ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 476 | . . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → 𝑒 = 𝐸) | |
2 | dmeq 5246 | . . . 4 ⊢ (𝑒 = 𝐸 → dom 𝑒 = dom 𝐸) | |
3 | 2 | adantl 481 | . . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → dom 𝑒 = dom 𝐸) |
4 | pweq 4111 | . . . . 5 ⊢ (𝑣 = 𝑉 → 𝒫 𝑣 = 𝒫 𝑉) | |
5 | 4 | difeq1d 3689 | . . . 4 ⊢ (𝑣 = 𝑉 → (𝒫 𝑣 ∖ {∅}) = (𝒫 𝑉 ∖ {∅})) |
6 | 5 | adantr 480 | . . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝒫 𝑣 ∖ {∅}) = (𝒫 𝑉 ∖ {∅})) |
7 | 1, 3, 6 | feq123d 5947 | . 2 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅}) ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
8 | df-uhgra 25821 | . 2 ⊢ UHGrph = {〈𝑣, 𝑒〉 ∣ 𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})} | |
9 | 7, 8 | brabga 4914 | 1 ⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝑉 UHGrph 𝐸 ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 ∅c0 3874 𝒫 cpw 4108 {csn 4125 class class class wbr 4583 dom cdm 5038 ⟶wf 5800 UHGrph cuhg 25819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 df-uhgra 25821 |
This theorem is referenced by: uhgraf 25828 ushgrauhgra 25832 uhgraop 25833 uhgraeq12d 25836 uhgrares 25837 uhgra0 25838 uhgra0v 25839 uhgraun 25840 umisuhgra 25856 uhgruhgra 40375 |
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