Theorem isubgruhgr 48225

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 29147	. . . . . 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 5978	. . . . . 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 6200	. . . . . 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 7035	. . . . . . . . . . . . . . . . 17 ⊢ (((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}) ∧ 𝑦 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑉 ∖ {∅}))
9		eldifsni 4748	. . . . . . . . . . . . . . . . 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 6857	. . . . . . . . . . . . 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 4562	. . . . . . . . . . . 12 ⊢ (((iEdg‘𝐺)‘𝑦) ⊆ 𝑆 → ((iEdg‘𝐺)‘𝑦) ∈ 𝒫 𝑆)
18	14, 17	anim12ci 615	. . . . . . . . . . 11 ⊢ ((((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) ∧ 𝑦 ∈ dom (iEdg‘𝐺)) ∧ ((iEdg‘𝐺)‘𝑦) ⊆ 𝑆) → (((iEdg‘𝐺)‘𝑦) ∈ 𝒫 𝑆 ∧ ((iEdg‘𝐺)‘𝑦) ≠ ∅))
19		eldifsn 4744	. . . . . . . . . . 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 3130	. . . . . . . 8 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → ∀𝑦 ∈ dom (iEdg‘𝐺)(((iEdg‘𝐺)‘𝑦) ⊆ 𝑆 → ((iEdg‘𝐺)‘𝑦) ∈ (𝒫 𝑆 ∖ {∅})))
23		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((iEdg‘𝐺)‘𝑥) = ((iEdg‘𝐺)‘𝑦))
24	23	sseq1d 3967	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((iEdg‘𝐺)‘𝑥) ⊆ 𝑆 ↔ ((iEdg‘𝐺)‘𝑦) ⊆ 𝑆))
25	24	ralrab 3654	. . . . . . . 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 6673	. . . . . . . . . . 11 ⊢ ((iEdg‘𝐺):dom (iEdg‘𝐺)⟶(𝒫 𝑉 ∖ {∅}) → Fun (iEdg‘𝐺))
28		ssrab2 4034	. . . . . . . . . . 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 6906	. . . . . . . 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 3974	. . . . 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 7081	. . . 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 6198	. . . . . 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 6661	. . . 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 48220	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (iEdg‘(𝐺 ISubGr 𝑆)) = ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
42	41	dmeqd 5862	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → dom (iEdg‘(𝐺 ISubGr 𝑆)) = dom ((iEdg‘𝐺) ↾ {𝑥 ∈ dom (iEdg‘𝐺) ∣ ((iEdg‘𝐺)‘𝑥) ⊆ 𝑆}))
43	1	isubgrvtx 48224	. . . . . 6 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (Vtx‘(𝐺 ISubGr 𝑆)) = 𝑆)
44	43	pweqd 4573	. . . . 5 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → 𝒫 (Vtx‘(𝐺 ISubGr 𝑆)) = 𝒫 𝑆)
45	44	difeq1d 4079	. . . 4 ⊢ ((𝐺 ∈ UHGraph ∧ 𝑆 ⊆ 𝑉) → (𝒫 (Vtx‘(𝐺 ISubGr 𝑆)) ∖ {∅}) = (𝒫 𝑆 ∖ {∅}))
46	41, 42, 45	feq123d 6659	. . 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 7403	. . 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 29145	. . 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 1542   ∈ wcel 2114   ≠ wne 2933  ∀wral 3052  {crab 3401  Vcvv 3442   ∖ cdif 3900   ⊆ wss 3903  ∅c0 4287  𝒫 cpw 4556  {csn 4582  dom cdm 5632   ↾ cres 5634   “ cima 5635  Fun wfun 6494  ⟶wf 6496  ‘cfv 6500  (class class class)co 7368  Vtxcvtx 29081  iEdgciedg 29082  UHGraphcuhgr 29141   ISubGr cisubgr 48217

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-vtx 29083  df-iedg 29084  df-uhgr 29143  df-isubgr 48218

This theorem is referenced by:  isubgrsubgr  48226  grlicref  48369  grlicsym  48370  clnbgr3stgrgrlim  48376  clnbgr3stgrgrlic  48377