Theorem isubgrvtxuhgr 48026

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 3954	. . 3 ⊢ (𝐺 ∈ UHGraph → 𝑉 ⊆ 𝑉)
2		isubgriedg.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3		isubgriedg.e	. . . 4 ⊢ 𝐸 = (iEdg‘𝐺)
4	2, 3	isisubgr 48024	. . 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 29065	. . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
7		funrel 6506	. . . . 5 ⊢ (Fun 𝐸 → Rel 𝐸)
8	6, 7	syl 17	. . . 4 ⊢ (𝐺 ∈ UHGraph → Rel 𝐸)
9	2, 3	uhgrf 29061	. . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
10		ffvelcdm 7023	. . . . . . . 8 ⊢ ((𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑥 ∈ dom 𝐸) → (𝐸‘𝑥) ∈ (𝒫 𝑉 ∖ {∅}))
11		eldifi 4080	. . . . . . . . 9 ⊢ ((𝐸‘𝑥) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸‘𝑥) ∈ 𝒫 𝑉)
12	11	elpwid 4560	. . . . . . . 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 3407	. . . . . 6 ⊢ (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) → {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉} = dom 𝐸)
15	14	eqimsscd 3988	. . . . 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 5978	. . . 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 4833	. 2 ⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})⟩ = ⟨𝑉, 𝐸⟩)
20	5, 19	eqtrd 2768	1 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = ⟨𝑉, 𝐸⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3396   ∖ cdif 3895   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4551  {csn 4577  ⟨cop 4583  dom cdm 5621   ↾ cres 5623  Rel wrel 5626  Fun wfun 6483  ⟶wf 6485  ‘cfv 6489  (class class class)co 7355  Vtxcvtx 28995  iEdgciedg 28996  UHGraphcuhgr 29055   ISubGr cisubgr 48022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497  df-ov 7358  df-oprab 7359  df-mpo 7360  df-uhgr 29057  df-isubgr 48023

This theorem is referenced by: (None)