Theorem isisubgr 47892

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 3457	. . 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 6836	. . . . 5 ⊢ 𝑉 ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝑆 ⊆ 𝑉 → 𝑉 ∈ V)
6		id 22	. . . 4 ⊢ (𝑆 ⊆ 𝑉 → 𝑆 ⊆ 𝑉)
7	5, 6	sselpwd 5266	. . 3 ⊢ (𝑆 ⊆ 𝑉 → 𝑆 ∈ 𝒫 𝑉)
8	7	adantl 481	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 𝑆 ∈ 𝒫 𝑉)
9		opex 5404	. . 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 6837	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → (iEdg‘𝑔) ∈ V)
13		fveq2 6822	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
14		isisubgr.e	. . . . . . . . . 10 ⊢ 𝐸 = (iEdg‘𝐺)
15	13, 14	eqtr4di 2784	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐸)
16	15	eqeq2d 2742	. . . . . . . 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 5843	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝐸 → dom 𝑒 = dom 𝐸)
20	19	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → dom 𝑒 = dom 𝐸)
21		fveq1 6821	. . . . . . . . . . . . 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 3968	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → ((𝑒‘𝑥) ⊆ 𝑣 ↔ (𝐸‘𝑥) ⊆ 𝑆))
25	20, 24	rabeqbidv 3413	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣} = {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})
26	18, 25	reseq12d 5929	. . . . . . . . 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 3886	. . . 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 4833	. . 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 6822	. . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
34	33, 3	eqtr4di 2784	. . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
35	34	pweqd 4567	. . 3 ⊢ (𝑔 = 𝐺 → 𝒫 (Vtx‘𝑔) = 𝒫 𝑉)
36		df-isubgr 47891	. . 3 ⊢ ISubGr = (𝑔 ∈ V, 𝑣 ∈ 𝒫 (Vtx‘𝑔) ↦ ⟨𝑣, ⦋(iEdg‘𝑔) / 𝑒⦌(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣})⟩)
37	32, 35, 36	ovmpox 7499	. 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 1541   ∈ wcel 2111  {crab 3395  Vcvv 3436  ⦋csb 3850   ⊆ wss 3902  𝒫 cpw 4550  ⟨cop 4582  dom cdm 5616   ↾ cres 5618  ‘cfv 6481  (class class class)co 7346  Vtxcvtx 28972  iEdgciedg 28973   ISubGr cisubgr 47890

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-res 5628  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-isubgr 47891

This theorem is referenced by:  isubgriedg  47893  isubgrvtxuhgr  47894  isubgrvtx  47897  isubgr0uhgr  47903