Theorem isubgruhgr 48114

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 2736	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	uhgrf 29135	. . . . . 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 5970	. . . . . 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 6192	. . . . . 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 7026	. . . . . . . . . . . . . . . . 17 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑦 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑉 ∖ {∅}))
9		eldifsni 4746	. . . . . . . . . . . . . . . . 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 6849	. . . . . . . . . . . . 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 4560	. . . . . . . . . . . 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 4742	. . . . . . . . . . 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 3128	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ∀𝑦 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑦) ⊆ 𝑆 → ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑆 ∖ {∅})))
23		fveq2 6834	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝐺)‘𝑦))
24	23	sseq1d 3965	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((iEdg‘𝐺)‘𝑥) ⊆ 𝑆 ↔ ((iEdg‘𝐺)‘𝑦) ⊆ 𝑆))
25	24	ralrab 3652	. . . . . . . 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 6665	. . . . . . . . . . 11 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}) → Fun (iEdg‘𝐺))
28		ssrab2 4032	. . . . . . . . . . 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 6898	. . . . . . . 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 3972	. . . . 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 7071	. . . 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 6190	. . . . . 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 2745	. . . . 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 6653	. . . 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 48109	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
42	41	dmeqd 5854	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → dom (iEdg‘(𝐺 ISubGr 𝑆)) = dom ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
43	1	isubgrvtx 48113	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) = 𝑆)
44	43	pweqd 4571	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → 𝒫 (Vtx‘(𝐺 ISubGr 𝑆)) = 𝒫 𝑆)
45	44	difeq1d 4077	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝒫 (Vtx‘(𝐺 ISubGr 𝑆)) ∖ {∅}) = (𝒫 𝑆 ∖ {∅}))
46	41, 42, 45	feq123d 6651	. . 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 7393	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ V)
49		eqid 2736	. . . 4 ⊢ (Vtx‘(𝐺 ISubGr 𝑆)) = (Vtx‘(𝐺 ISubGr 𝑆))
50		eqid 2736	. . . 4 ⊢ (iEdg‘(𝐺 ISubGr 𝑆)) = (iEdg‘(𝐺 ISubGr 𝑆))
51	49, 50	isuhgr 29133	. . 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 1541   ∈ wcel 2113   ≠ wne 2932  ∀wral 3051  {crab 3399  Vcvv 3440   ∖ cdif 3898   ⊆ wss 3901  ∅c0 4285  𝒫 cpw 4554  {csn 4580  dom cdm 5624   ↾ cres 5626   “ cima 5627  Fun wfun 6486  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  Vtxcvtx 29069  iEdgciedg 29070  UHGraphcuhgr 29129   ISubGr cisubgr 48106

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  ax-un 7680

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-mpt 5180  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-ima 5637  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-1st 7933  df-2nd 7934  df-vtx 29071  df-iedg 29072  df-uhgr 29131  df-isubgr 48107

This theorem is referenced by:  isubgrsubgr  48115  grlicref  48258  grlicsym  48259  clnbgr3stgrgrlim  48265  clnbgr3stgrgrlic  48266