Theorem isubgrvtxuhgr 48780

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 48778	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ⊆ 𝑉) → (𝐺 ISubGr 𝑉) = ⟨𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})⟩)
5	1, 4	mpdan 700	. 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = ⟨𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})⟩)
6	3	uhgrfun 29523	. . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
7		funrel 6550	. . . . 5 ⊢ (Fun 𝐸 → Rel 𝐸)
8	6, 7	syl 18	. . . 4 ⊢ (𝐺 ∈ UHGraph → Rel 𝐸)
9	2, 3	uhgrf 29519	. . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
10		ffvelcdm 7074	. . . . . . . 8 ⊢ ((𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑥 ∈ dom 𝐸) → (𝐸‘𝑥) ∈ (𝒫 𝑉 ∖ {∅}))
11		eldifi 4078	. . . . . . . . 9 ⊢ ((𝐸‘𝑥) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸‘𝑥) ∈ 𝒫 𝑉)
12	11	elpwid 4566	. . . . . . . 8 ⊢ ((𝐸‘𝑥) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸‘𝑥) ⊆ 𝑉)
13	10, 12	syl 18	. . . . . . 7 ⊢ ((𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑥 ∈ dom 𝐸) → (𝐸‘𝑥) ⊆ 𝑉)
14	13	rabeqcda 3423	. . . . . 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 18	. . . 4 ⊢ (𝐺 ∈ UHGraph → dom 𝐸 ⊆ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})
17		relssres 6015	. . . 4 ⊢ ((Rel 𝐸 ∧ dom 𝐸 ⊆ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉}) → (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉}) = 𝐸)
18	8, 16, 17	syl2anc 596	. . 3 ⊢ (𝐺 ∈ UHGraph → (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉}) = 𝐸)
19	18	opeq2d 4840	. 2 ⊢ (𝐺 ∈ UHGraph → ⟨𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})⟩ = ⟨𝑉, 𝐸⟩)
20	5, 19	eqtrd 2795	1 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = ⟨𝑉, 𝐸⟩)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  {crab 3412   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  ⟨cop 4590  dom cdm 5655   ↾ cres 5657  Rel wrel 5660  Fun wfun 6527  ⟶wf 6529  ‘cfv 6533  (class class class)co 7413  Vtxcvtx 29453  iEdgciedg 29454  UHGraphcuhgr 29513   ISubGr cisubgr 48776

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7416  df-oprab 7417  df-mpo 7418  df-uhgr 29515  df-isubgr 48777

This theorem is used by: (None)