Theorem isubgrvtxuhgr 47325

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 4000	. . 3 ⊢ (𝐺 ∈ UHGraph → 𝑉 ⊆ 𝑉)
2		isubgriedg.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
3		isubgriedg.e	. . . 4 ⊢ 𝐸 = (iEdg‘𝐺)
4	2, 3	isisubgr 47323	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑉 ⊆ 𝑉) → (𝐺 ISubGr 𝑉) = ⟨𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})⟩)
5	1, 4	mpdan 685	. 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr 𝑉) = ⟨𝑉, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉})⟩)
6	3	uhgrfun 28950	. . . . 5 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
7		funrel 6571	. . . . 5 ⊢ (Fun 𝐸 → Rel 𝐸)
8	6, 7	syl 17	. . . 4 ⊢ (𝐺 ∈ UHGraph → Rel 𝐸)
9	2, 3	uhgrf 28946	. . . . 5 ⊢ (𝐺 ∈ UHGraph → 𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}))
10		ffvelcdm 7090	. . . . . . . 8 ⊢ ((𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑥 ∈ dom 𝐸) → (𝐸‘𝑥) ∈ (𝒫 𝑉 ∖ {∅}))
11		eldifi 4123	. . . . . . . . 9 ⊢ ((𝐸‘𝑥) ∈ (𝒫 𝑉 ∖ {∅}) → (𝐸‘𝑥) ∈ 𝒫 𝑉)
12	11	elpwid 4613	. . . . . . . 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 3430	. . . . . 6 ⊢ (𝐸:dom 𝐸⟶(𝒫 𝑉 ∖ {∅}) → {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉} = dom 𝐸)
15	14	eqimsscd 4034	. . . . 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 6027	. . . 4 ⊢ ((Rel 𝐸 ∧ dom 𝐸 ⊆ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉}) → (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉}) = 𝐸)
18	8, 16, 17	syl2anc 582	. . 3 ⊢ (𝐺 ∈ UHGraph → (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑉}) = 𝐸)
19	18	opeq2d 4882	. 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 394   = wceq 1533   ∈ wcel 2098  {crab 3418   ∖ cdif 3941   ⊆ wss 3944  ∅c0 4322  𝒫 cpw 4604  {csn 4630  ⟨cop 4636  dom cdm 5678   ↾ cres 5680  Rel wrel 5683  Fun wfun 6543  ⟶wf 6545  ‘cfv 6549  (class class class)co 7419  Vtxcvtx 28880  iEdgciedg 28881  UHGraphcuhgr 28940   ISubGr cisubgr 47321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-iota 6501  df-fun 6551  df-fn 6552  df-f 6553  df-fv 6557  df-ov 7422  df-oprab 7423  df-mpo 7424  df-uhgr 28942  df-isubgr 47322

This theorem is referenced by: (None)

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