Theorem isubgruhgr 48910

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 2761	. . . . . . 7 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
3	1, 2	uhgrf 29622	. . . . . 6 ⊢ (𝐺 ∈ UHGraph → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
4	3	adantr 486	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}))
5		dmresss 6002	. . . . . 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 6228	. . . . . 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 7073	. . . . . . . . . . . . . . . . 17 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑦 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑉 ∖ {∅}))
9		eldifsni 4753	. . . . . . . . . . . . . . . . 17 ⊢ (((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑉 ∖ {∅}) → ((iEdg‘𝐺)‘𝑦) ≠ ∅)
10	8, 9	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑦 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑦) ≠ ∅)
11	10	ex 418	. . . . . . . . . . . . . . 15 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}) → (𝑦 ∈ dom (iEdg‘𝐺) → ((iEdg‘𝐺)‘𝑦) ≠ ∅))
12	3, 11	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝐺 ∈ UHGraph → (𝑦 ∈ dom (iEdg‘𝐺) → ((iEdg‘𝐺)‘𝑦) ≠ ∅))
13	12	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝑦 ∈ dom (iEdg‘𝐺) → ((iEdg‘𝐺)‘𝑦) ≠ ∅))
14	13	imp 412	. . . . . . . . . . . 12 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝑦 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑦) ≠ ∅)
15		fvexd 6892	. . . . . . . . . . . . 13 ⊢ (((iEdg‘𝐺)‘𝑦) ⊆ 𝑆 → ((iEdg‘𝐺)‘𝑦) ∈ V)
16		id 23	. . . . . . . . . . . . 13 ⊢ (((iEdg‘𝐺)‘𝑦) ⊆ 𝑆 → ((iEdg‘𝐺)‘𝑦) ⊆ 𝑆)
17	15, 16	elpwd 4563	. . . . . . . . . . . 12 ⊢ (((iEdg‘𝐺)‘𝑦) ⊆ 𝑆 → ((iEdg‘𝐺)‘𝑦) ∈ 𝒫 𝑆)
18	14, 17	anim12ci 626	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝑦 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐺)‘𝑦) ⊆ 𝑆) → (((iEdg‘𝐺)‘𝑦) ∈ 𝒫 𝑆 ∧ ((iEdg‘𝐺)‘𝑦) ≠ ∅))
19		eldifsn 4748	. . . . . . . . . . 11 ⊢ (((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑆 ∖ {∅}) ↔ (((iEdg‘𝐺)‘𝑦) ∈ 𝒫 𝑆 ∧ ((iEdg‘𝐺)‘𝑦) ≠ ∅))
20	18, 19	sylibr 237	. . . . . . . . . 10 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝑦 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐺)‘𝑦) ⊆ 𝑆) → ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑆 ∖ {∅}))
21	20	ex 418	. . . . . . . . 9 ⊢ (((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝑦 ∈ dom (iEdg‘𝐺)) → (((iEdg‘𝐺)‘𝑦) ⊆ 𝑆 → ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑆 ∖ {∅})))
22	21	ralrimiva 3155	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ∀𝑦 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑦) ⊆ 𝑆 → ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑆 ∖ {∅})))
23		fveq2 6877	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝐺)‘𝑦))
24	23	sseq1d 3962	. . . . . . . . 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 237	. . . . . . 7 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ∀𝑦 ∈ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆} ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑆 ∖ {∅}))
27		ffun 6704	. . . . . . . . . . 11 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}) → Fun (iEdg‘𝐺))
28		ssrab2 4028	. . . . . . . . . . 11 ⊢ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆} ⊆ dom (iEdg‘𝐺)
29	27, 28	jctir 530	. . . . . . . . . 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 18	. . . . . . . . 9 ⊢ (𝐺 ∈ UHGraph → (Fun (iEdg‘𝐺) ∧ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆} ⊆ dom (iEdg‘𝐺)))
31	30	adantr 486	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (Fun (iEdg‘𝐺) ∧ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆} ⊆ dom (iEdg‘𝐺)))
32		funimass4 6941	. . . . . . . 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 18	. . . . . . 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 260	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ((iEdg‘𝐺) “ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}) ⊆ (𝒫 𝑆 ∖ {∅}))
35	7, 34	eqsstrid 3969	. . . . 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 7119	. . . 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 6226	. . . . . 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 2770	. . . . 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 6692	. . . 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 237	. . 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 48905	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
42	41	dmeqd 5887	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → dom (iEdg‘(𝐺 ISubGr 𝑆)) = dom ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
43	1	isubgrvtx 48909	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) = 𝑆)
44	43	pweqd 4574	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → 𝒫 (Vtx‘(𝐺 ISubGr 𝑆)) = 𝒫 𝑆)
45	44	difeq1d 4073	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝒫 (Vtx‘(𝐺 ISubGr 𝑆)) ∖ {∅}) = (𝒫 𝑆 ∖ {∅}))
46	41, 42, 45	feq123d 6690	. . 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 260	. 2 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)):dom (iEdg‘(𝐺 ISubGr 𝑆))⟶(𝒫 (Vtx‘(𝐺 ISubGr 𝑆)) ∖ {∅}))
48		ovexd 7447	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ V)
49		eqid 2761	. . . 4 ⊢ (Vtx‘(𝐺 ISubGr 𝑆)) = (Vtx‘(𝐺 ISubGr 𝑆))
50		eqid 2761	. . . 4 ⊢ (iEdg‘(𝐺 ISubGr 𝑆)) = (iEdg‘(𝐺 ISubGr 𝑆))
51	49, 50	isuhgr 29620	. . 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 18	. 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 260	1 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) ∈ UHGraph)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  {crab 3413  Vcvv 3451   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  𝒫 cpw 4557  {csn 4584  dom cdm 5651   ↾ cres 5653   “ cima 5654  Fun wfun 6525  ⟶wf 6527  ‘cfv 6531  (class class class)co 7412  Vtxcvtx 29556  iEdgciedg 29557  UHGraphcuhgr 29616   ISubGr cisubgr 48902

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 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7740

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  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-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7990  df-2nd 7991  df-vtx 29558  df-iedg 29559  df-uhgr 29618  df-isubgr 48903

This theorem is used by:  isubgrsubgr  48911  grlicref  49054  grlicsym  49055  clnbgr3stgrgrlim  49061  clnbgr3stgrgrlic  49062