Theorem isubgr0uhgr 48115

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 4352	. . 3 ⊢ ∅ ⊆ (Vtx‘𝐺)
2		eqid 2736	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2736	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	2, 3	isisubgr 48104	. . 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 4269	. . . . . . 7 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} ∩ dom (iEdg‘𝐺)) = {𝑥 ∈ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}
7		inidm 4179	. . . . . . . . 9 ⊢ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) = dom (iEdg‘𝐺)
8	7	rabeqi 3412	. . . . . . . 8 ⊢ {𝑥 ∈ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}
9		ss0b 4353	. . . . . . . 8 ⊢ (((iEdg‘𝐺)‘𝑥) ⊆ ∅ ↔ ((iEdg‘𝐺)‘𝑥) = ∅)
10	8, 9	rabbieq 3407	. . . . . . 7 ⊢ {𝑥 ∈ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅}
11	6, 10	eqtri 2759	. . . . . 6 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} ∩ dom (iEdg‘𝐺)) = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅}
12	11	ineqcomi 4163	. . . . 5 ⊢ (dom (iEdg‘𝐺) ∩ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅}
13	2, 3	uhgrf 29135	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
14		ffvelcdm 7026	. . . . . . . . . 10 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑥) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
15		eldifsni 4746	. . . . . . . . . 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 3128	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ ((iEdg‘𝐺)‘𝑥) = ∅)
20		rabeq0 4340	. . . . . 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 2783	. . . 4 ⊢ (𝐺 ∈ UHGraph → (dom (iEdg‘𝐺) ∩ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = ∅)
23	3	uhgrfun 29139	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
24	23	funfnd 6523	. . . . 5 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
25		fnresdisj 6612	. . . . 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 4836	. 2 ⊢ (𝐺 ∈ UHGraph → ⟨∅, ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅})⟩ = ⟨∅, ∅⟩)
29	5, 28	eqtrd 2771	1 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr ∅) = ⟨∅, ∅⟩)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  {crab 3399   ∖ cdif 3898   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  {csn 4580  ⟨cop 4586  dom cdm 5624   ↾ cres 5626   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  Vtxcvtx 29069  iEdgciedg 29070  UHGraphcuhgr 29129   ISubGr cisubgr 48102

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 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  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 7361  df-oprab 7362  df-mpo 7363  df-uhgr 29131  df-isubgr 48103

This theorem is referenced by: (None)