Theorem isubgr0uhgr 47998

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 4349	. . 3 ⊢ ∅ ⊆ (Vtx‘𝐺)
2		eqid 2733	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2733	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	2, 3	isisubgr 47987	. . 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 4266	. . . . . . 7 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} ∩ dom (iEdg‘𝐺)) = {𝑥 ∈ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}
7		inidm 4176	. . . . . . . . 9 ⊢ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) = dom (iEdg‘𝐺)
8	7	rabeqi 3409	. . . . . . . 8 ⊢ {𝑥 ∈ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}
9		ss0b 4350	. . . . . . . 8 ⊢ (((iEdg‘𝐺)‘𝑥) ⊆ ∅ ↔ ((iEdg‘𝐺)‘𝑥) = ∅)
10	8, 9	rabbieq 3404	. . . . . . 7 ⊢ {𝑥 ∈ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅}
11	6, 10	eqtri 2756	. . . . . 6 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} ∩ dom (iEdg‘𝐺)) = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅}
12	11	ineqcomi 4160	. . . . 5 ⊢ (dom (iEdg‘𝐺) ∩ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅}
13	2, 3	uhgrf 29042	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
14		ffvelcdm 7020	. . . . . . . . . 10 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑥) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
15		eldifsni 4741	. . . . . . . . . 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 2934	. . . . . . . 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 3125	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ ((iEdg‘𝐺)‘𝑥) = ∅)
20		rabeq0 4337	. . . . . 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 2780	. . . 4 ⊢ (𝐺 ∈ UHGraph → (dom (iEdg‘𝐺) ∩ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = ∅)
23	3	uhgrfun 29046	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
24	23	funfnd 6517	. . . . 5 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
25		fnresdisj 6606	. . . . 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 4831	. 2 ⊢ (𝐺 ∈ UHGraph → ⟨∅, ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅})⟩ = ⟨∅, ∅⟩)
29	5, 28	eqtrd 2768	1 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr ∅) = ⟨∅, ∅⟩)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2929  ∀wral 3048  {crab 3396   ∖ cdif 3895   ∩ cin 3897   ⊆ wss 3898  ∅c0 4282  𝒫 cpw 4549  {csn 4575  ⟨cop 4581  dom cdm 5619   ↾ cres 5621   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352  Vtxcvtx 28976  iEdgciedg 28977  UHGraphcuhgr 29036   ISubGr cisubgr 47985

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 5236  ax-nul 5246  ax-pr 5372

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 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-uhgr 29038  df-isubgr 47986

This theorem is referenced by: (None)