Theorem isisubgr 49175

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 3475	. . 3 ⊢ (𝐺 ∈ 𝑊 → 𝐺 ∈ V)
2	1	adantr 486	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 𝐺 ∈ V)
3		isisubgr.v	. . . . . 6 ⊢ 𝑉 = (Vtx‘𝐺)
4	3	fvexi 6925	. . . . 5 ⊢ 𝑉 ∈ V
5	4	a1i 11	. . . 4 ⊢ (𝑆 ⊆ 𝑉 → 𝑉 ∈ V)
6		id 23	. . . 4 ⊢ (𝑆 ⊆ 𝑉 → 𝑆 ⊆ 𝑉)
7	5, 6	sselpwd 5295	. . 3 ⊢ (𝑆 ⊆ 𝑉 → 𝑆 ∈ 𝒫 𝑉)
8	7	adantl 487	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → 𝑆 ∈ 𝒫 𝑉)
9		opex 5437	. . 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 490	. . . 4 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → 𝑣 = 𝑆)
12		fvexd 6926	. . . . 5 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → (iEdg‘𝑔) ∈ V)
13		fveq2 6911	. . . . . . . . . 10 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺))
14		isisubgr.e	. . . . . . . . . 10 ⊢ 𝐸 = (iEdg‘𝐺)
15	13, 14	eqtr4di 2816	. . . . . . . . 9 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = 𝐸)
16	15	eqeq2d 2774	. . . . . . . 8 ⊢ (𝑔 = 𝐺 → (𝑒 = (iEdg‘𝑔) ↔ 𝑒 = 𝐸))
17	16	adantr 486	. . . . . . 7 ⊢ ((𝑔 = 𝐺 ∧ 𝑣 = 𝑆) → (𝑒 = (iEdg‘𝑔) ↔ 𝑒 = 𝐸))
18		simpr 490	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → 𝑒 = 𝐸)
19		dmeq 5893	. . . . . . . . . . . 12 ⊢ (𝑒 = 𝐸 → dom 𝑒 = dom 𝐸)
20	19	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → dom 𝑒 = dom 𝐸)
21		fveq1 6910	. . . . . . . . . . . . 13 ⊢ (𝑒 = 𝐸 → (𝑒‘𝑥) = (𝐸‘𝑥))
22	21	adantl 487	. . . . . . . . . . . 12 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → (𝑒‘𝑥) = (𝐸‘𝑥))
23		simpl 488	. . . . . . . . . . . 12 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → 𝑣 = 𝑆)
24	22, 23	sseq12d 3946	. . . . . . . . . . 11 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → ((𝑒‘𝑥) ⊆ 𝑣 ↔ (𝐸‘𝑥) ⊆ 𝑆))
25	20, 24	rabeqbidv 3433	. . . . . . . . . 10 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣} = {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})
26	18, 25	reseq12d 5983	. . . . . . . . 9 ⊢ ((𝑣 = 𝑆 ∧ 𝑒 = 𝐸) → (𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆}))
27	26	ex 418	. . . . . . . 8 ⊢ (𝑣 = 𝑆 → (𝑒 = 𝐸 → (𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣}) = (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})))
28	27	adantl 487	. . . . . . 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 243	. . . . . 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 412	. . . . 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 3887	. . . 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 4851	. . 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 6911	. . . . 5 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺))
34	33, 3	eqtr4di 2816	. . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉)
35	34	pweqd 4577	. . 3 ⊢ (𝑔 = 𝐺 → 𝒫 (Vtx‘𝑔) = 𝒫 𝑉)
36		df-isubgr 49174	. . 3 ⊢ ISubGr = (𝑔 ∈ V, 𝑣 ∈ 𝒫 (Vtx‘𝑔) ↦ ⟨𝑣, ⦋(iEdg‘𝑔) / 𝑒⦌(𝑒 ↾ {𝑥 ∈ dom 𝑒 ∣ (𝑒‘𝑥) ⊆ 𝑣})⟩)
37	32, 35, 36	ovmpox 7600	. 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 1399	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ⊆ 𝑉) → (𝐺 ISubGr 𝑆) = ⟨𝑆, (𝐸 ↾ {𝑥 ∈ dom 𝐸 ∣ (𝐸‘𝑥) ⊆ 𝑆})⟩)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454  ⦋csb 3851   ⊆ wss 3900  𝒫 cpw 4560  ⟨cop 4838  dom cdm 5656   ↾ cres 5658  ‘cfv 6561  (class class class)co 7446  Vtxcvtx 29607  iEdgciedg 29608   ISubGr cisubgr 49173

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-res 5668  df-iota 6515  df-fun 6563  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-isubgr 49174

This theorem is used by:  isubgriedg  49176  isubgrvtxuhgr  49177  isubgrvtx  49180  isubgr0uhgr  49186