Theorem isubgr0uhgr 48789

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 4350	. . 3 ⊢ ∅ ⊆ (Vtx‘𝐺)
2		eqid 2760	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
3		eqid 2760	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
4	2, 3	isisubgr 48778	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ ∅ ⊆ (Vtx‘𝐺)) → (𝐺 ISubGr ∅) = ⟨∅, ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅})⟩)
5	1, 4	mpan2 704	. 2 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr ∅) = ⟨∅, ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅})⟩)
6		inrab2 4263	. . . . . . 7 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} ∩ dom (iEdg‘𝐺)) = {𝑥 ∈ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}
7		inidm 4172	. . . . . . . . 9 ⊢ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) = dom (iEdg‘𝐺)
8	7	rabeqi 3425	. . . . . . . 8 ⊢ {𝑥 ∈ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}
9		ss0b 4351	. . . . . . . 8 ⊢ (((iEdg‘𝐺)‘𝑥) ⊆ ∅ ↔ ((iEdg‘𝐺)‘𝑥) = ∅)
10	8, 9	rabbieq 3420	. . . . . . 7 ⊢ {𝑥 ∈ (dom (iEdg‘𝐺) ∩ dom (iEdg‘𝐺)) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅}
11	6, 10	eqtri 2783	. . . . . 6 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅} ∩ dom (iEdg‘𝐺)) = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅}
12	11	ineqcomi 4157	. . . . 5 ⊢ (dom (iEdg‘𝐺) ∩ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅}
13	2, 3	uhgrf 29519	. . . . . . . 8 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}))
14		ffvelcdm 7074	. . . . . . . . . 10 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑥) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}))
15		eldifsni 4753	. . . . . . . . . 10 ⊢ (((iEdg‘𝐺)‘𝑥) ∈ (𝒫 (Vtx‘𝐺) ∖ {∅}) → ((iEdg‘𝐺)‘𝑥) ≠ ∅)
16	14, 15	syl 18	. . . . . . . . 9 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑥) ≠ ∅)
17	16	neneqd 2960	. . . . . . . 8 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 (Vtx‘𝐺) ∖ {∅}) ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → ¬ ((iEdg‘𝐺)‘𝑥) = ∅)
18	13, 17	sylan 592	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑥 ∈ dom (iEdg‘𝐺)) → ¬ ((iEdg‘𝐺)‘𝑥) = ∅)
19	18	ralrimiva 3154	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ ((iEdg‘𝐺)‘𝑥) = ∅)
20		rabeq0 4338	. . . . . 6 ⊢ ({𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅} = ∅ ↔ ∀𝑥 ∈ dom (iEdg‘𝐺) ¬ ((iEdg‘𝐺)‘𝑥) = ∅)
21	19, 20	sylibr 237	. . . . 5 ⊢ (𝐺 ∈ UHGraph → {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) = ∅} = ∅)
22	12, 21	eqtrid 2807	. . . 4 ⊢ (𝐺 ∈ UHGraph → (dom (iEdg‘𝐺) ∩ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = ∅)
23	3	uhgrfun 29523	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → Fun (iEdg‘𝐺))
24	23	funfnd 6564	. . . . 5 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺) Fn dom (iEdg‘𝐺))
25		fnresdisj 6652	. . . . 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 18	. . . 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 235	. . 3 ⊢ (𝐺 ∈ UHGraph → ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅}) = ∅)
28	27	opeq2d 4840	. 2 ⊢ (𝐺 ∈ UHGraph → ⟨∅, ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ ∅})⟩ = ⟨∅, ∅⟩)
29	5, 28	eqtrd 2795	1 ⊢ (𝐺 ∈ UHGraph → (𝐺 ISubGr ∅) = ⟨∅, ∅⟩)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2955  ∀wral 3076  {crab 3412   ∖ cdif 3896   ∩ cin 3898   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  ⟨cop 4590  dom cdm 5655   ↾ cres 5657   Fn wfn 6528  ⟶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)