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Theorem isuhgr 26235
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 26233 . . 3 UHGraph = {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})}
21eleq2i 2836 . 2 (𝐺 ∈ UHGraph ↔ 𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})})
3 fveq2 6377 . . . . 5 ( = 𝐺 → (iEdg‘) = (iEdg‘𝐺))
4 isuhgr.e . . . . 5 𝐸 = (iEdg‘𝐺)
53, 4syl6eqr 2817 . . . 4 ( = 𝐺 → (iEdg‘) = 𝐸)
63dmeqd 5496 . . . . 5 ( = 𝐺 → dom (iEdg‘) = dom (iEdg‘𝐺))
74eqcomi 2774 . . . . . 6 (iEdg‘𝐺) = 𝐸
87dmeqi 5495 . . . . 5 dom (iEdg‘𝐺) = dom 𝐸
96, 8syl6eq 2815 . . . 4 ( = 𝐺 → dom (iEdg‘) = dom 𝐸)
10 fveq2 6377 . . . . . . 7 ( = 𝐺 → (Vtx‘) = (Vtx‘𝐺))
11 isuhgr.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
1210, 11syl6eqr 2817 . . . . . 6 ( = 𝐺 → (Vtx‘) = 𝑉)
1312pweqd 4322 . . . . 5 ( = 𝐺 → 𝒫 (Vtx‘) = 𝒫 𝑉)
1413difeq1d 3891 . . . 4 ( = 𝐺 → (𝒫 (Vtx‘) ∖ {∅}) = (𝒫 𝑉 ∖ {∅}))
155, 9, 14feq123d 6214 . . 3 ( = 𝐺 → ((iEdg‘):dom (iEdg‘)⟶(𝒫 (Vtx‘) ∖ {∅}) ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
16 fvexd 6392 . . . . 5 (𝑔 = → (Vtx‘𝑔) ∈ V)
17 fveq2 6377 . . . . 5 (𝑔 = → (Vtx‘𝑔) = (Vtx‘))
18 fvexd 6392 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) ∈ V)
19 fveq2 6377 . . . . . . 7 (𝑔 = → (iEdg‘𝑔) = (iEdg‘))
2019adantr 472 . . . . . 6 ((𝑔 = 𝑣 = (Vtx‘)) → (iEdg‘𝑔) = (iEdg‘))
21 simpr 477 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → 𝑒 = (iEdg‘))
2221dmeqd 5496 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → dom 𝑒 = dom (iEdg‘))
23 simpr 477 . . . . . . . . . 10 ((𝑔 = 𝑣 = (Vtx‘)) → 𝑣 = (Vtx‘))
2423pweqd 4322 . . . . . . . . 9 ((𝑔 = 𝑣 = (Vtx‘)) → 𝒫 𝑣 = 𝒫 (Vtx‘))
2524difeq1d 3891 . . . . . . . 8 ((𝑔 = 𝑣 = (Vtx‘)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘) ∖ {∅}))
2625adantr 472 . . . . . . 7 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝒫 𝑣 ∖ {∅}) = (𝒫 (Vtx‘) ∖ {∅}))
2721, 22, 26feq123d 6214 . . . . . 6 (((𝑔 = 𝑣 = (Vtx‘)) ∧ 𝑒 = (iEdg‘)) → (𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘):dom (iEdg‘)⟶(𝒫 (Vtx‘) ∖ {∅})))
2818, 20, 27sbcied2 3636 . . . . 5 ((𝑔 = 𝑣 = (Vtx‘)) → ([(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘):dom (iEdg‘)⟶(𝒫 (Vtx‘) ∖ {∅})))
2916, 17, 28sbcied2 3636 . . . 4 (𝑔 = → ([(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅}) ↔ (iEdg‘):dom (iEdg‘)⟶(𝒫 (Vtx‘) ∖ {∅})))
3029cbvabv 2890 . . 3 {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})} = { ∣ (iEdg‘):dom (iEdg‘)⟶(𝒫 (Vtx‘) ∖ {∅})}
3115, 30elab2g 3510 . 2 (𝐺𝑈 → (𝐺 ∈ {𝑔[(Vtx‘𝑔) / 𝑣][(iEdg‘𝑔) / 𝑒]𝑒:dom 𝑒⟶(𝒫 𝑣 ∖ {∅})} ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
322, 31syl5bb 274 1 (𝐺𝑈 → (𝐺 ∈ UHGraph ↔ 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wcel 2155  {cab 2751  Vcvv 3350  [wsbc 3598  cdif 3731  c0 4081  𝒫 cpw 4317  {csn 4336  dom cdm 5279  wf 6066  cfv 6070  Vtxcvtx 26168  iEdgciedg 26169  UHGraphcuhgr 26231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-nul 4951
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-fv 6078  df-uhgr 26233
This theorem is referenced by:  uhgrf  26237  uhgreq12g  26240  ushgruhgr  26244  isuhgrop  26245  uhgr0e  26246  uhgr0  26248  uhgrun  26249  uhgrstrrepe  26253  incistruhgr  26254  upgruhgr  26277  subuhgr  26460
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