Theorem isubgrvtxuhgr 48352

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 3946	. . 3 ⊢ (𝐺 ∈ UHGraph → 𝑉 ⊆ 𝑉)
2		isubgriedg.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3		isubgriedg.e	. . . 4 ⊢ 𝐸 = (iEdg‘𝐺)
4	2, 3	isisubgr 48350	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ⊆ 𝑉) → (𝐺 ISubGr 𝑉) = ⟨𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})⟩)
5	1, 4	mpdan 688	. 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = ⟨𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})⟩)
6	3	uhgrfun 29149	. . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
7		funrel 6509	. . . . 5 ⊢ (Fun 𝐸 → Rel 𝐸)
8	6, 7	syl 17	. . . 4 ⊢ (𝐺 ∈ UHGraph → Rel 𝐸)
9	2, 3	uhgrf 29145	. . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
10		ffvelcdm 7027	. . . . . . . 8 ⊢ ((𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑥 ∈ dom 𝐸) → (𝐸‘𝑥) ∈ (𝒫 𝑉 ∖ {∅}))
11		eldifi 4072	. . . . . . . . 9 ⊢ ((𝐸‘𝑥) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸‘𝑥) ∈ 𝒫 𝑉)
12	11	elpwid 4551	. . . . . . . 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 3401	. . . . . 6 ⊢ (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) → {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉} = dom 𝐸)
15	14	eqimsscd 3980	. . . . 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 5981	. . . 4 ⊢ ((Rel 𝐸 ∧ dom 𝐸 ⊆ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉}) → (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉}) = 𝐸)
18	8, 16, 17	syl2anc 585	. . 3 ⊢ (𝐺 ∈ UHGraph → (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉}) = 𝐸)
19	18	opeq2d 4824	. 2 ⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})⟩ = ⟨𝑉, 𝐸⟩)
20	5, 19	eqtrd 2772	1 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = ⟨𝑉, 𝐸⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {crab 3390   ∖ cdif 3887   ⊆ wss 3890  ∅c0 4274  𝒫 cpw 4542  {csn 4568  ⟨cop 4574  dom cdm 5624   ↾ cres 5626  Rel wrel 5629  Fun wfun 6486  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  Vtxcvtx 29079  iEdgciedg 29080  UHGraphcuhgr 29139   ISubGr cisubgr 48348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-uhgr 29141  df-isubgr 48349

This theorem is referenced by: (None)