Theorem isubgruhgr 47854

Description: An induced subgraph of a hypergraph is a hypergraph. (Contributed by AV, 13-May-2025.)

Hypothesis

Ref	Expression
isubgrvtx.v	⊢ 𝑉 = (Vtx‘𝐺)

Assertion

Ref	Expression
isubgruhgr	⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UHGraph)

Proof of Theorem isubgruhgr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isubgrvtx.v	. . . . . . 7 ⊢ 𝑉 = (Vtx‘𝐺)
2		eqid 2737	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	uhgrf 29079	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
4	3	adantr 480	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
5		dmresss 6029	. . . . . 6 ⊢ dom ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ dom (iEdg‘𝐺)
6	5	a1i 11	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → dom ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ dom (iEdg‘𝐺))
7		imadmres 6254	. . . . . 6 ⊢ ((iEdg‘𝐺) “ dom ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})) = ((iEdg‘𝐺) “ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})
8		ffvelcdm 7101	. . . . . . . . . . . . . . . . 17 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑦 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑉 ∖ {∅}))
9		eldifsni 4790	. . . . . . . . . . . . . . . . 17 ⊢ (((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑉 ∖ {∅}) → ((iEdg‘𝐺)‘𝑦) ≠ ∅)
10	8, 9	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑦 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑦) ≠ ∅)
11	10	ex 412	. . . . . . . . . . . . . . 15 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}) → (𝑦 ∈ dom (iEdg‘𝐺) → ((iEdg‘𝐺)‘𝑦) ≠ ∅))
12	3, 11	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ UHGraph → (𝑦 ∈ dom (iEdg‘𝐺) → ((iEdg‘𝐺)‘𝑦) ≠ ∅))
13	12	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝑦 ∈ dom (iEdg‘𝐺) → ((iEdg‘𝐺)‘𝑦) ≠ ∅))
14	13	imp 406	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝑦 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑦) ≠ ∅)
15		fvexd 6921	. . . . . . . . . . . . 13 ⊢ (((iEdg‘𝐺)‘𝑦) ⊆ 𝑆 → ((iEdg‘𝐺)‘𝑦) ∈ V)
16		id 22	. . . . . . . . . . . . 13 ⊢ (((iEdg‘𝐺)‘𝑦) ⊆ 𝑆 → ((iEdg‘𝐺)‘𝑦) ⊆ 𝑆)
17	15, 16	elpwd 4606	. . . . . . . . . . . 12 ⊢ (((iEdg‘𝐺)‘𝑦) ⊆ 𝑆 → ((iEdg‘𝐺)‘𝑦) ∈ 𝒫 𝑆)
18	14, 17	anim12ci 614	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝑦 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐺)‘𝑦) ⊆ 𝑆) → (((iEdg‘𝐺)‘𝑦) ∈ 𝒫 𝑆 ∧ ((iEdg‘𝐺)‘𝑦) ≠ ∅))
19		eldifsn 4786	. . . . . . . . . . 11 ⊢ (((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑆 ∖ {∅}) ↔ (((iEdg‘𝐺)‘𝑦) ∈ 𝒫 𝑆 ∧ ((iEdg‘𝐺)‘𝑦) ≠ ∅))
20	18, 19	sylibr 234	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝑦 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐺)‘𝑦) ⊆ 𝑆) → ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑆 ∖ {∅}))
21	20	ex 412	. . . . . . . . 9 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝑦 ∈ dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘𝑦) ⊆ 𝑆 → ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑆 ∖ {∅})))
22	21	ralrimiva 3146	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ∀𝑦 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑦) ⊆ 𝑆 → ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑆 ∖ {∅})))
23		fveq2 6906	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝐺)‘𝑦))
24	23	sseq1d 4015	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((iEdg‘𝐺)‘𝑥) ⊆ 𝑆 ↔ ((iEdg‘𝐺)‘𝑦) ⊆ 𝑆))
25	24	ralrab 3699	. . . . . . . 8 ⊢ (∀𝑦 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆} ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑆 ∖ {∅}) ↔ ∀𝑦 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑦) ⊆ 𝑆 → ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑆 ∖ {∅})))
26	22, 25	sylibr 234	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ∀𝑦 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆} ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑆 ∖ {∅}))
27		ffun 6739	. . . . . . . . . . 11 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}) → Fun (iEdg‘𝐺))
28		ssrab2 4080	. . . . . . . . . . 11 ⊢ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆} ⊆ dom (iEdg‘𝐺)
29	27, 28	jctir 520	. . . . . . . . . 10 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}) → (Fun (iEdg‘𝐺) ∧ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆} ⊆ dom (iEdg‘𝐺)))
30	3, 29	syl 17	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → (Fun (iEdg‘𝐺) ∧ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆} ⊆ dom (iEdg‘𝐺)))
31	30	adantr 480	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (Fun (iEdg‘𝐺) ∧ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆} ⊆ dom (iEdg‘𝐺)))
32		funimass4 6973	. . . . . . . 8 ⊢ ((Fun (iEdg‘𝐺) ∧ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆} ⊆ dom (iEdg‘𝐺)) → (((iEdg‘𝐺) “ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (𝒫 𝑆 ∖ {∅}) ↔ ∀𝑦 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆} ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑆 ∖ {∅})))
33	31, 32	syl 17	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (((iEdg‘𝐺) “ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (𝒫 𝑆 ∖ {∅}) ↔ ∀𝑦 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆} ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑆 ∖ {∅})))
34	26, 33	mpbird 257	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ((iEdg‘𝐺) “ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (𝒫 𝑆 ∖ {∅}))
35	7, 34	eqsstrid 4022	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ((iEdg‘𝐺) “ dom ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})) ⊆ (𝒫 𝑆 ∖ {∅}))
36	4, 6, 35	fssrescdmd 7146	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ((iEdg‘𝐺) ↾ dom ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})):dom ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})⟶(𝒫 𝑆 ∖ {∅}))
37		resdmres 6252	. . . . . 6 ⊢ ((iEdg‘𝐺) ↾ dom ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})
38	37	eqcomi 2746	. . . . 5 ⊢ ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) = ((iEdg‘𝐺) ↾ dom ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
39	38	feq1i 6727	. . . 4 ⊢ (((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}):dom ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})⟶(𝒫 𝑆 ∖ {∅}) ↔ ((iEdg‘𝐺) ↾ dom ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})):dom ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})⟶(𝒫 𝑆 ∖ {∅}))
40	36, 39	sylibr 234	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}):dom ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})⟶(𝒫 𝑆 ∖ {∅}))
41	1, 2	isubgriedg 47849	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
42	41	dmeqd 5916	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → dom (iEdg‘(𝐺 ISubGr 𝑆)) = dom ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
43	1	isubgrvtx 47853	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) = 𝑆)
44	43	pweqd 4617	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → 𝒫 (Vtx‘(𝐺 ISubGr 𝑆)) = 𝒫 𝑆)
45	44	difeq1d 4125	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝒫 (Vtx‘(𝐺 ISubGr 𝑆)) ∖ {∅}) = (𝒫 𝑆 ∖ {∅}))
46	41, 42, 45	feq123d 6725	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ((iEdg‘(𝐺 ISubGr 𝑆)):dom (iEdg‘(𝐺 ISubGr 𝑆))⟶(𝒫 (Vtx‘(𝐺 ISubGr 𝑆)) ∖ {∅}) ↔ ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}):dom ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆})⟶(𝒫 𝑆 ∖ {∅})))
47	40, 46	mpbird 257	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)):dom (iEdg‘(𝐺 ISubGr 𝑆))⟶(𝒫 (Vtx‘(𝐺 ISubGr 𝑆)) ∖ {∅}))
48		ovexd 7466	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ V)
49		eqid 2737	. . . 4 ⊢ (Vtx‘(𝐺 ISubGr 𝑆)) = (Vtx‘(𝐺 ISubGr 𝑆))
50		eqid 2737	. . . 4 ⊢ (iEdg‘(𝐺 ISubGr 𝑆)) = (iEdg‘(𝐺 ISubGr 𝑆))
51	49, 50	isuhgr 29077	. . 3 ⊢ ((𝐺 ISubGr 𝑆) ∈ V → ((𝐺 ISubGr 𝑆) ∈ UHGraph ↔ (iEdg‘(𝐺 ISubGr 𝑆)):dom (iEdg‘(𝐺 ISubGr 𝑆))⟶(𝒫 (Vtx‘(𝐺 ISubGr 𝑆)) ∖ {∅})))
52	48, 51	syl 17	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ((𝐺 ISubGr 𝑆) ∈ UHGraph ↔ (iEdg‘(𝐺 ISubGr 𝑆)):dom (iEdg‘(𝐺 ISubGr 𝑆))⟶(𝒫 (Vtx‘(𝐺 ISubGr 𝑆)) ∖ {∅})))
53	47, 52	mpbird 257	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UHGraph)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  {crab 3436  Vcvv 3480   ∖ cdif 3948   ⊆ wss 3951  ∅c0 4333  𝒫 cpw 4600  {csn 4626  dom cdm 5685   ↾ cres 5687   “ cima 5688  Fun wfun 6555  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  Vtxcvtx 29013  iEdgciedg 29014  UHGraphcuhgr 29073   ISubGr cisubgr 47846

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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-vtx 29015  df-iedg 29016  df-uhgr 29075  df-isubgr 47847

This theorem is referenced by:  isubgrsubgr  47855  grlicref  47972  grlicsym  47973  clnbgr3stgrgrlic  47979