Theorem isisubgr 47734

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 3509	. . 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 6934	. . . . 5 ⊢ 𝑉 ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝑆 ⊆ 𝑉 → 𝑉 ∈ V)
6		id 22	. . . 4 ⊢ (𝑆 ⊆ 𝑉 → 𝑆 ⊆ 𝑉)
7	5, 6	sselpwd 5346	. . 3 ⊢ (𝑆 ⊆ 𝑉 → 𝑆 ∈ 𝒫 𝑉)
8	7	adantl 481	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 𝑆 ∈ 𝒫 𝑉)
9		opex 5484	. . 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 6935	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → (iEdg‘𝑔) ∈ V)
13		fveq2 6920	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
14		isisubgr.e	. . . . . . . . . 10 ⊢ 𝐸 = (iEdg‘𝐺)
15	13, 14	eqtr4di 2798	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐸)
16	15	eqeq2d 2751	. . . . . . . 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 5928	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝐸 → dom 𝑒 = dom 𝐸)
20	19	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → dom 𝑒 = dom 𝐸)
21		fveq1 6919	. . . . . . . . . . . . 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 4042	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → ((𝑒‘𝑥) ⊆ 𝑣 ↔ (𝐸‘𝑥) ⊆ 𝑆))
25	20, 24	rabeqbidv 3462	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣} = {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})
26	18, 25	reseq12d 6010	. . . . . . . . 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 3959	. . . 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 4905	. . 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 6920	. . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
34	33, 3	eqtr4di 2798	. . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
35	34	pweqd 4639	. . 3 ⊢ (𝑔 = 𝐺 → 𝒫 (Vtx‘𝑔) = 𝒫 𝑉)
36		df-isubgr 47733	. . 3 ⊢ ISubGr = (𝑔 ∈ V, 𝑣 ∈ 𝒫 (Vtx‘𝑔) ↦ ⟨𝑣, ⦋(iEdg‘𝑔) / 𝑒⦌(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣})⟩)
37	32, 35, 36	ovmpox 7603	. 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 1371	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 1537   ∈ wcel 2108  {crab 3443  Vcvv 3488  ⦋csb 3921   ⊆ wss 3976  𝒫 cpw 4622  ⟨cop 4654  dom cdm 5700   ↾ cres 5702  ‘cfv 6573  (class class class)co 7448  Vtxcvtx 29031  iEdgciedg 29032   ISubGr cisubgr 47732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-isubgr 47733

This theorem is referenced by:  isubgriedg  47735  isubgrvtxuhgr  47736  isubgrvtx  47737  isubgr0uhgr  47743