Theorem isubgrvtxuhgr 47868

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.

Step	Hyp	Ref	Expression
1		ssidd 3973	. . 3 ⊢ (𝐺 ∈ UHGraph → 𝑉 ⊆ 𝑉)
2		isubgriedg.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3		isubgriedg.e	. . . 4 ⊢ 𝐸 = (iEdg‘𝐺)
4	2, 3	isisubgr 47866	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ⊆ 𝑉) → (𝐺 ISubGr 𝑉) = ⟨𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})⟩)
5	1, 4	mpdan 687	. 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = ⟨𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})⟩)
6	3	uhgrfun 29000	. . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
7		funrel 6536	. . . . 5 ⊢ (Fun 𝐸 → Rel 𝐸)
8	6, 7	syl 17	. . . 4 ⊢ (𝐺 ∈ UHGraph → Rel 𝐸)
9	2, 3	uhgrf 28996	. . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
10		ffvelcdm 7056	. . . . . . . 8 ⊢ ((𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑥 ∈ dom 𝐸) → (𝐸‘𝑥) ∈ (𝒫 𝑉 ∖ {∅}))
11		eldifi 4097	. . . . . . . . 9 ⊢ ((𝐸‘𝑥) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸‘𝑥) ∈ 𝒫 𝑉)
12	11	elpwid 4575	. . . . . . . 8 ⊢ ((𝐸‘𝑥) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸‘𝑥) ⊆ 𝑉)
13	10, 12	syl 17	. . . . . . 7 ⊢ ((𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑥 ∈ dom 𝐸) → (𝐸‘𝑥) ⊆ 𝑉)
14	13	rabeqcda 3420	. . . . . 6 ⊢ (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) → {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉} = dom 𝐸)
15	14	eqimsscd 4007	. . . . 5 ⊢ (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) → dom 𝐸 ⊆ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})
16	9, 15	syl 17	. . . 4 ⊢ (𝐺 ∈ UHGraph → dom 𝐸 ⊆ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})
17		relssres 5996	. . . 4 ⊢ ((Rel 𝐸 ∧ dom 𝐸 ⊆ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉}) → (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉}) = 𝐸)
18	8, 16, 17	syl2anc 584	. . 3 ⊢ (𝐺 ∈ UHGraph → (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉}) = 𝐸)
19	18	opeq2d 4847	. 2 ⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})⟩ = ⟨𝑉, 𝐸⟩)
20	5, 19	eqtrd 2765	1 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = ⟨𝑉, 𝐸⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3408   ∖ cdif 3914   ⊆ wss 3917  ∅c0 4299  𝒫 cpw 4566  {csn 4592  ⟨cop 4598  dom cdm 5641   ↾ cres 5643  Rel wrel 5646  Fun wfun 6508  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  Vtxcvtx 28930  iEdgciedg 28931  UHGraphcuhgr 28990   ISubGr cisubgr 47864

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-uhgr 28992  df-isubgr 47865

This theorem is referenced by: (None)