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Theorem isuhgr 28011
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 28009 . . 3 UHGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}
21eleq2i 2829 . 2 (𝐺 ∈ UHGraph ↔ 𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})})
3 fveq2 6842 . . . . 5 ( = 𝐺 → (iEdg‘) = (iEdg‘𝐺))
4 isuhgr.e . . . . 5 𝐸 = (iEdg‘𝐺)
53, 4eqtr4di 2794 . . . 4 ( = 𝐺 → (iEdg‘) = 𝐸)
63dmeqd 5861 . . . . 5 ( = 𝐺 → dom (iEdg‘) = dom (iEdg‘𝐺))
74eqcomi 2745 . . . . . 6 (iEdg‘𝐺) = 𝐸
87dmeqi 5860 . . . . 5 dom (iEdg‘𝐺) = dom 𝐸
96, 8eqtrdi 2792 . . . 4 ( = 𝐺 → dom (iEdg‘) = dom 𝐸)
10 fveq2 6842 . . . . . . 7 ( = 𝐺 → (Vtx‘) = (Vtx‘𝐺))
11 isuhgr.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
1210, 11eqtr4di 2794 . . . . . 6 ( = 𝐺 → (Vtx‘) = 𝑉)
1312pweqd 4577 . . . . 5 ( = 𝐺 → 𝒫 (Vtx‘) = 𝒫 𝑉)
1413difeq1d 4081 . . . 4 ( = 𝐺 → (𝒫 (Vtx‘) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
155, 9, 14feq123d 6657 . . 3 ( = 𝐺 → ((iEdg‘):dom (iEdg‘)⟶(𝒫 (Vtx‘) ∖ {∅}) ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
16 fvexd 6857 . . . . 5 (𝑔 = → (Vtx‘𝑔) ∈ V)
17 fveq2 6842 . . . . 5 (𝑔 = → (Vtx‘𝑔) = (Vtx‘))
18 fvexd 6857 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) ∈ V)
19 fveq2 6842 . . . . . . 7 (𝑔 = → (iEdg‘𝑔) = (iEdg‘))
2019adantr 481 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) = (iEdg‘))
21 simpr 485 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → 𝑒 = (iEdg‘))
2221dmeqd 5861 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → dom 𝑒 = dom (iEdg‘))
23 simpr 485 . . . . . . . . . 10 ((𝑔 = 𝑣 = (Vtx‘)) → 𝑣 = (Vtx‘))
2423pweqd 4577 . . . . . . . . 9 ((𝑔 = 𝑣 = (Vtx‘)) → 𝒫 𝑣 = 𝒫 (Vtx‘))
2524difeq1d 4081 . . . . . . . 8 ((𝑔 = 𝑣 = (Vtx‘)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘) ∖ {∅}))
2625adantr 481 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘) ∖ {∅}))
2721, 22, 26feq123d 6657 . . . . . 6 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘):dom (iEdg‘)⟶(𝒫 (Vtx‘) ∖ {∅})))
2818, 20, 27sbcied2 3786 . . . . 5 ((𝑔 = 𝑣 = (Vtx‘)) → ([(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘):dom (iEdg‘)⟶(𝒫 (Vtx‘) ∖ {∅})))
2916, 17, 28sbcied2 3786 . . . 4 (𝑔 = → ([(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘):dom (iEdg‘)⟶(𝒫 (Vtx‘) ∖ {∅})))
3029cbvabv 2809 . . 3 {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})} = { ∣ (iEdg‘):dom (iEdg‘)⟶(𝒫 (Vtx‘) ∖ {∅})}
3115, 30elab2g 3632 . 2 (𝐺𝑈 → (𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})} ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
322, 31bitrid 282 1 (𝐺𝑈 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  {cab 2713  Vcvv 3445  [wsbc 3739  cdif 3907  c0 4282  𝒫 cpw 4560  {csn 4586  dom cdm 5633  wf 6492  cfv 6496  Vtxcvtx 27947  iEdgciedg 27948  UHGraphcuhgr 28007
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5263
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fv 6504  df-uhgr 28009
This theorem is referenced by:  uhgrf  28013  uhgreq12g  28016  ushgruhgr  28020  isuhgrop  28021  uhgr0e  28022  uhgr0  28024  uhgrun  28025  uhgrstrrepe  28029  incistruhgr  28030  upgruhgr  28053  subuhgr  28234
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