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Theorem isuhgr 29077
Description: The predicate "is an undirected hypergraph." (Contributed by Alexander van der Vekens, 26-Dec-2017.) (Revised by AV, 9-Oct-2020.)
Hypotheses
Ref Expression
isuhgr.v 𝑉 = (Vtx‘𝐺)
isuhgr.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
isuhgr (𝐺𝑈 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))

Proof of Theorem isuhgr
Dummy variables 𝑔 𝑣 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-uhgr 29075 . . 3 UHGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}
21eleq2i 2833 . 2 (𝐺 ∈ UHGraph ↔ 𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})})
3 fveq2 6906 . . . . 5 ( = 𝐺 → (iEdg‘) = (iEdg‘𝐺))
4 isuhgr.e . . . . 5 𝐸 = (iEdg‘𝐺)
53, 4eqtr4di 2795 . . . 4 ( = 𝐺 → (iEdg‘) = 𝐸)
63dmeqd 5916 . . . . 5 ( = 𝐺 → dom (iEdg‘) = dom (iEdg‘𝐺))
74eqcomi 2746 . . . . . 6 (iEdg‘𝐺) = 𝐸
87dmeqi 5915 . . . . 5 dom (iEdg‘𝐺) = dom 𝐸
96, 8eqtrdi 2793 . . . 4 ( = 𝐺 → dom (iEdg‘) = dom 𝐸)
10 fveq2 6906 . . . . . . 7 ( = 𝐺 → (Vtx‘) = (Vtx‘𝐺))
11 isuhgr.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
1210, 11eqtr4di 2795 . . . . . 6 ( = 𝐺 → (Vtx‘) = 𝑉)
1312pweqd 4617 . . . . 5 ( = 𝐺 → 𝒫 (Vtx‘) = 𝒫 𝑉)
1413difeq1d 4125 . . . 4 ( = 𝐺 → (𝒫 (Vtx‘) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
155, 9, 14feq123d 6725 . . 3 ( = 𝐺 → ((iEdg‘):dom (iEdg‘)⟶(𝒫 (Vtx‘) ∖ {∅}) ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
16 fvexd 6921 . . . . 5 (𝑔 = → (Vtx‘𝑔) ∈ V)
17 fveq2 6906 . . . . 5 (𝑔 = → (Vtx‘𝑔) = (Vtx‘))
18 fvexd 6921 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) ∈ V)
19 fveq2 6906 . . . . . . 7 (𝑔 = → (iEdg‘𝑔) = (iEdg‘))
2019adantr 480 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) = (iEdg‘))
21 simpr 484 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → 𝑒 = (iEdg‘))
2221dmeqd 5916 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → dom 𝑒 = dom (iEdg‘))
23 simpr 484 . . . . . . . . . 10 ((𝑔 = 𝑣 = (Vtx‘)) → 𝑣 = (Vtx‘))
2423pweqd 4617 . . . . . . . . 9 ((𝑔 = 𝑣 = (Vtx‘)) → 𝒫 𝑣 = 𝒫 (Vtx‘))
2524difeq1d 4125 . . . . . . . 8 ((𝑔 = 𝑣 = (Vtx‘)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘) ∖ {∅}))
2625adantr 480 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘) ∖ {∅}))
2721, 22, 26feq123d 6725 . . . . . 6 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘):dom (iEdg‘)⟶(𝒫 (Vtx‘) ∖ {∅})))
2818, 20, 27sbcied2 3833 . . . . 5 ((𝑔 = 𝑣 = (Vtx‘)) → ([(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘):dom (iEdg‘)⟶(𝒫 (Vtx‘) ∖ {∅})))
2916, 17, 28sbcied2 3833 . . . 4 (𝑔 = → ([(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘):dom (iEdg‘)⟶(𝒫 (Vtx‘) ∖ {∅})))
3029cbvabv 2812 . . 3 {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})} = { ∣ (iEdg‘):dom (iEdg‘)⟶(𝒫 (Vtx‘) ∖ {∅})}
3115, 30elab2g 3680 . 2 (𝐺𝑈 → (𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})} ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
322, 31bitrid 283 1 (𝐺𝑈 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {cab 2714  Vcvv 3480  [wsbc 3788  cdif 3948  c0 4333  𝒫 cpw 4600  {csn 4626  dom cdm 5685  wf 6557  cfv 6561  Vtxcvtx 29013  iEdgciedg 29014  UHGraphcuhgr 29073
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-uhgr 29075
This theorem is referenced by:  uhgrf  29079  uhgreq12g  29082  ushgruhgr  29086  isuhgrop  29087  uhgr0e  29088  uhgr0  29090  uhgrun  29091  uhgrstrrepe  29095  incistruhgr  29096  upgruhgr  29119  subuhgr  29303  isubgruhgr  47854  grimuhgr  47878
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