Theorem isisubgr 47875

Description: The subgraph induced by a subset of vertices. (Contributed by AV, 12-May-2025.)

Hypotheses

Ref	Expression
isisubgr.v	⊢ 𝑉 = (Vtx‘𝐺)
isisubgr.e	⊢ 𝐸 = (iEdg‘𝐺)

Assertion

Ref	Expression
isisubgr	⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) = ⟨𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})⟩)

Distinct variable groups:   𝑥,𝐸   𝑥,𝐺   𝑥,𝑆   𝑥,𝑉

Allowed substitution hint:   𝑊(𝑥)

Proof of Theorem isisubgr

Dummy variables 𝑒 𝑔 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3480	. . 3 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
2	1	adantr 480	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 𝐺 ∈ V)
3		isisubgr.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
4	3	fvexi 6890	. . . . 5 ⊢ 𝑉 ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝑆 ⊆ 𝑉 → 𝑉 ∈ V)
6		id 22	. . . 4 ⊢ (𝑆 ⊆ 𝑉 → 𝑆 ⊆ 𝑉)
7	5, 6	sselpwd 5298	. . 3 ⊢ (𝑆 ⊆ 𝑉 → 𝑆 ∈ 𝒫 𝑉)
8	7	adantl 481	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 𝑆 ∈ 𝒫 𝑉)
9		opex 5439	. . 3 ⊢ ⟨𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})⟩ ∈ V
10	9	a1i 11	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → ⟨𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})⟩ ∈ V)
11		simpr 484	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → 𝑣 = 𝑆)
12		fvexd 6891	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → (iEdg‘𝑔) ∈ V)
13		fveq2 6876	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
14		isisubgr.e	. . . . . . . . . 10 ⊢ 𝐸 = (iEdg‘𝐺)
15	13, 14	eqtr4di 2788	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐸)
16	15	eqeq2d 2746	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑒 = (iEdg‘𝑔) ↔ 𝑒 = 𝐸))
17	16	adantr 480	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → (𝑒 = (iEdg‘𝑔) ↔ 𝑒 = 𝐸))
18		simpr 484	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → 𝑒 = 𝐸)
19		dmeq 5883	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝐸 → dom 𝑒 = dom 𝐸)
20	19	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → dom 𝑒 = dom 𝐸)
21		fveq1 6875	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝐸 → (𝑒‘𝑥) = (𝐸‘𝑥))
22	21	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → (𝑒‘𝑥) = (𝐸‘𝑥))
23		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → 𝑣 = 𝑆)
24	22, 23	sseq12d 3992	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → ((𝑒‘𝑥) ⊆ 𝑣 ↔ (𝐸‘𝑥) ⊆ 𝑆))
25	20, 24	rabeqbidv 3434	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣} = {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})
26	18, 25	reseq12d 5967	. . . . . . . . 9 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → (𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆}))
27	26	ex 412	. . . . . . . 8 ⊢ (𝑣 = 𝑆 → (𝑒 = 𝐸 → (𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})))
28	27	adantl 481	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → (𝑒 = 𝐸 → (𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})))
29	17, 28	sylbid 240	. . . . . 6 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → (𝑒 = (iEdg‘𝑔) → (𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})))
30	29	imp 406	. . . . 5 ⊢ (((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) ∧ 𝑒 = (iEdg‘𝑔)) → (𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆}))
31	12, 30	csbied 3910	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → ⦋(iEdg‘𝑔) / 𝑒⦌(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆}))
32	11, 31	opeq12d 4857	. . 3 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → ⟨𝑣, ⦋(iEdg‘𝑔) / 𝑒⦌(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣})⟩ = ⟨𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})⟩)
33		fveq2 6876	. . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
34	33, 3	eqtr4di 2788	. . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
35	34	pweqd 4592	. . 3 ⊢ (𝑔 = 𝐺 → 𝒫 (Vtx‘𝑔) = 𝒫 𝑉)
36		df-isubgr 47874	. . 3 ⊢ ISubGr = (𝑔 ∈ V, 𝑣 ∈ 𝒫 (Vtx‘𝑔) ↦ ⟨𝑣, ⦋(iEdg‘𝑔) / 𝑒⦌(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣})⟩)
37	32, 35, 36	ovmpox 7560	. 2 ⊢ ((𝐺 ∈ V ∧ 𝑆 ∈ 𝒫 𝑉 ∧ ⟨𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})⟩ ∈ V) → (𝐺 ISubGr 𝑆) = ⟨𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})⟩)
38	2, 8, 10, 37	syl3anc 1373	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) = ⟨𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})⟩)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2108  {crab 3415  Vcvv 3459  ⦋csb 3874   ⊆ wss 3926  𝒫 cpw 4575  ⟨cop 4607  dom cdm 5654   ↾ cres 5656  ‘cfv 6531  (class class class)co 7405  Vtxcvtx 28975  iEdgciedg 28976   ISubGr cisubgr 47873

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-res 5666  df-iota 6484  df-fun 6533  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-isubgr 47874

This theorem is referenced by:  isubgriedg  47876  isubgrvtxuhgr  47877  isubgrvtx  47880  isubgr0uhgr  47886