Theorem isubgr0uhgr 47886

Description: The subgraph induced by an empty set of vertices of a hypergraph. (Contributed by AV, 13-May-2025.)

Assertion

Ref	Expression
isubgr0uhgr	⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr ∅) = ⟨∅, ∅⟩)

Proof of Theorem isubgr0uhgr

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ss 4375	. . 3 ⊢ ∅ ⊆ (Vtx‘𝐺)
2		eqid 2735	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2735	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	2, 3	isisubgr 47875	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ∅ ⊆ (Vtx‘𝐺)) → (𝐺 ISubGr ∅) = ⟨∅, ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅})⟩)
5	1, 4	mpan2 691	. 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr ∅) = ⟨∅, ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅})⟩)
6		inrab2 4292	. . . . . . 7 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} ∩ dom (iEdg‘𝐺)) = {𝑥 ∈ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}
7		inidm 4202	. . . . . . . . 9 ⊢ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) = dom (iEdg‘𝐺)
8	7	rabeqi 3429	. . . . . . . 8 ⊢ {𝑥 ∈ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}
9		ss0b 4376	. . . . . . . 8 ⊢ (((iEdg‘𝐺)‘𝑥) ⊆ ∅ ↔ ((iEdg‘𝐺)‘𝑥) = ∅)
10	8, 9	rabbieq 3424	. . . . . . 7 ⊢ {𝑥 ∈ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅}
11	6, 10	eqtri 2758	. . . . . 6 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} ∩ dom (iEdg‘𝐺)) = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅}
12	11	ineqcomi 4186	. . . . 5 ⊢ (dom (iEdg‘𝐺) ∩ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅}
13	2, 3	uhgrf 29041	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
14		ffvelcdm 7071	. . . . . . . . . 10 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑥) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
15		eldifsni 4766	. . . . . . . . . 10 ⊢ (((iEdg‘𝐺)‘𝑥) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((iEdg‘𝐺)‘𝑥) ≠ ∅)
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑥) ≠ ∅)
17	16	neneqd 2937	. . . . . . . 8 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → ¬ ((iEdg‘𝐺)‘𝑥) = ∅)
18	13, 17	sylan 580	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → ¬ ((iEdg‘𝐺)‘𝑥) = ∅)
19	18	ralrimiva 3132	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ ((iEdg‘𝐺)‘𝑥) = ∅)
20		rabeq0 4363	. . . . . 6 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅} = ∅ ↔ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ ((iEdg‘𝐺)‘𝑥) = ∅)
21	19, 20	sylibr 234	. . . . 5 ⊢ (𝐺 ∈ UHGraph → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅} = ∅)
22	12, 21	eqtrid 2782	. . . 4 ⊢ (𝐺 ∈ UHGraph → (dom (iEdg‘𝐺) ∩ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = ∅)
23	3	uhgrfun 29045	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
24	23	funfnd 6567	. . . . 5 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
25		fnresdisj 6658	. . . . 5 ⊢ ((iEdg‘𝐺) Fn dom (iEdg‘𝐺) → ((dom (iEdg‘𝐺) ∩ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = ∅ ↔ ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = ∅))
26	24, 25	syl 17	. . . 4 ⊢ (𝐺 ∈ UHGraph → ((dom (iEdg‘𝐺) ∩ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = ∅ ↔ ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = ∅))
27	22, 26	mpbid 232	. . 3 ⊢ (𝐺 ∈ UHGraph → ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = ∅)
28	27	opeq2d 4856	. 2 ⊢ (𝐺 ∈ UHGraph → ⟨∅, ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅})⟩ = ⟨∅, ∅⟩)
29	5, 28	eqtrd 2770	1 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr ∅) = ⟨∅, ∅⟩)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2932  ∀wral 3051  {crab 3415   ∖ cdif 3923   ∩ cin 3925   ⊆ wss 3926  ∅c0 4308  𝒫 cpw 4575  {csn 4601  ⟨cop 4607  dom cdm 5654   ↾ cres 5656   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  (class class class)co 7405  Vtxcvtx 28975  iEdgciedg 28976  UHGraphcuhgr 29035   ISubGr cisubgr 47873

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-uhgr 29037  df-isubgr 47874

This theorem is referenced by: (None)