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Theorem isubgrvtxuhgr 47788
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 4019 . . 3 (𝐺 ∈ UHGraph → 𝑉𝑉)
2 isubgriedg.v . . . 4 𝑉 = (Vtx‘𝐺)
3 isubgriedg.e . . . 4 𝐸 = (iEdg‘𝐺)
42, 3isisubgr 47786 . . 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 29098 . . . . 5 (𝐺 ∈ UHGraph → Fun 𝐸)
7 funrel 6585 . . . . 5 (Fun 𝐸 → Rel 𝐸)
86, 7syl 17 . . . 4 (𝐺 ∈ UHGraph → Rel 𝐸)
92, 3uhgrf 29094 . . . . 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 4141 . . . . . . . . 9 ((𝐸𝑥) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸𝑥) ∈ 𝒫 𝑉)
1211elpwid 4614 . . . . . . . 8 ((𝐸𝑥) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸𝑥) ⊆ 𝑉)
1310, 12syl 17 . . . . . . 7 ((𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑥 ∈ dom 𝐸) → (𝐸𝑥) ⊆ 𝑉)
1413rabeqcda 3445 . . . . . 6 (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) → {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑉} = dom 𝐸)
1514eqimsscd 4053 . . . . 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 6042 . . . 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 4885 . 2 (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸𝑥) ⊆ 𝑉})⟩ = ⟨𝑉, 𝐸⟩)
205, 19eqtrd 2775 1 (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = ⟨𝑉, 𝐸⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  {crab 3433  cdif 3960  wss 3963  c0 4339  𝒫 cpw 4605  {csn 4631  cop 4637  dom cdm 5689  cres 5691  Rel wrel 5694  Fun wfun 6557  wf 6559  cfv 6563  (class class class)co 7431  Vtxcvtx 29028  iEdgciedg 29029  UHGraphcuhgr 29088   ISubGr cisubgr 47784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-uhgr 29090  df-isubgr 47785
This theorem is referenced by: (None)
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