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Theorem isubgrvtxuhgr 47850
Description: The subgraph induced by the full set of vertices of a hypergraph. (Contributed by AV, 12-May-2025.)
Hypotheses
Ref Expression
isubgriedg.v 𝑉 = (Vtx‘𝐺)
isubgriedg.e 𝐸 = (iEdg‘𝐺)
Assertion
Ref Expression
isubgrvtxuhgr (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = ⟨𝑉, 𝐸⟩)

Proof of Theorem isubgrvtxuhgr
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 ssidd 4007 . . 3 (𝐺 ∈ UHGraph → 𝑉𝑉)
2 isubgriedg.v . . . 4 𝑉 = (Vtx‘𝐺)
3 isubgriedg.e . . . 4 𝐸 = (iEdg‘𝐺)
42, 3isisubgr 47848 . . 3 ((𝐺 ∈ UHGraph ∧ 𝑉𝑉) → (𝐺 ISubGr 𝑉) = ⟨𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑉})⟩)
51, 4mpdan 687 . 2 (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = ⟨𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑉})⟩)
63uhgrfun 29083 . . . . 5 (𝐺 ∈ UHGraph → Fun 𝐸)
7 funrel 6583 . . . . 5 (Fun 𝐸 → Rel 𝐸)
86, 7syl 17 . . . 4 (𝐺 ∈ UHGraph → Rel 𝐸)
92, 3uhgrf 29079 . . . . 5 (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
10 ffvelcdm 7101 . . . . . . . 8 ((𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑥 ∈ dom 𝐸) → (𝐸𝑥) ∈ (𝒫 𝑉 ∖ {∅}))
11 eldifi 4131 . . . . . . . . 9 ((𝐸𝑥) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸𝑥) ∈ 𝒫 𝑉)
1211elpwid 4609 . . . . . . . 8 ((𝐸𝑥) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸𝑥) ⊆ 𝑉)
1310, 12syl 17 . . . . . . 7 ((𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑥 ∈ dom 𝐸) → (𝐸𝑥) ⊆ 𝑉)
1413rabeqcda 3448 . . . . . 6 (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) → {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑉} = dom 𝐸)
1514eqimsscd 4041 . . . . 5 (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) → dom 𝐸 ⊆ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑉})
169, 15syl 17 . . . 4 (𝐺 ∈ UHGraph → dom 𝐸 ⊆ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑉})
17 relssres 6040 . . . 4 ((Rel 𝐸 ∧ dom 𝐸 ⊆ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑉}) → (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑉}) = 𝐸)
188, 16, 17syl2anc 584 . . 3 (𝐺 ∈ UHGraph → (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑉}) = 𝐸)
1918opeq2d 4880 . 2 (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑉})⟩ = ⟨𝑉, 𝐸⟩)
205, 19eqtrd 2777 1 (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = ⟨𝑉, 𝐸⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  {crab 3436  cdif 3948  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626  cop 4632  dom cdm 5685  cres 5687  Rel wrel 5690  Fun wfun 6555  wf 6557  cfv 6561  (class class class)co 7431  Vtxcvtx 29013  iEdgciedg 29014  UHGraphcuhgr 29073   ISubGr cisubgr 47846
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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
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-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  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-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-uhgr 29075  df-isubgr 47847
This theorem is referenced by: (None)
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