Theorem isuhgra 25827

Description: The property of being an undirected hypergraph. (Contributed by Alexander van der Vekens, 26-Dec-2017.)

Assertion

Ref	Expression
isuhgra	⊢ ((𝑉 ∈ 𝑊 ∧ 𝐸 ∈ 𝑋) → (𝑉 UHGrph 𝐸 ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

Proof of Theorem isuhgra

Dummy variables 𝑣 𝑒 are mutually distinct and distinct from all other variables.

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 ∖ {∅})))

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