Theorem egrsubgr 26169

Description: An empty graph consisting of a subset of vertices of a graph (and having no edges) is a subgraph of the graph. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 17-Dec-2020.)

Assertion

Ref	Expression
egrsubgr	⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺)

Proof of Theorem egrsubgr

Step	Hyp	Ref	Expression
1		simp2 1062	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
2		eqid 2622	. . . . . . 7 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
3		eqid 2622	. . . . . . 7 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
4	2, 3	edg0iedg0 25949	. . . . . 6 ⊢ (Fun (iEdg‘𝑆) → ((Edg‘𝑆) = ∅ ↔ (iEdg‘𝑆) = ∅))
5	4	adantl 482	. . . . 5 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ Fun (iEdg‘𝑆)) → ((Edg‘𝑆) = ∅ ↔ (iEdg‘𝑆) = ∅))
6		res0 5400	. . . . . . 7 ⊢ ((iEdg‘𝐺) ↾ ∅) = ∅
7	6	eqcomi 2631	. . . . . 6 ⊢ ∅ = ((iEdg‘𝐺) ↾ ∅)
8		id 22	. . . . . 6 ⊢ ((iEdg‘𝑆) = ∅ → (iEdg‘𝑆) = ∅)
9		dmeq 5324	. . . . . . . 8 ⊢ ((iEdg‘𝑆) = ∅ → dom (iEdg‘𝑆) = dom ∅)
10		dm0 5339	. . . . . . . 8 ⊢ dom ∅ = ∅
11	9, 10	syl6eq 2672	. . . . . . 7 ⊢ ((iEdg‘𝑆) = ∅ → dom (iEdg‘𝑆) = ∅)
12	11	reseq2d 5396	. . . . . 6 ⊢ ((iEdg‘𝑆) = ∅ → ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) = ((iEdg‘𝐺) ↾ ∅))
13	7, 8, 12	3eqtr4a 2682	. . . . 5 ⊢ ((iEdg‘𝑆) = ∅ → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
14	5, 13	syl6bi 243	. . . 4 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ Fun (iEdg‘𝑆)) → ((Edg‘𝑆) = ∅ → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))))
15	14	impr 649	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
16	15	3adant2 1080	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
17		0ss 3972	. . . . 5 ⊢ ∅ ⊆ 𝒫 (Vtx‘𝑆)
18		sseq1 3626	. . . . 5 ⊢ ((Edg‘𝑆) = ∅ → ((Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆) ↔ ∅ ⊆ 𝒫 (Vtx‘𝑆)))
19	17, 18	mpbiri 248	. . . 4 ⊢ ((Edg‘𝑆) = ∅ → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
20	19	adantl 482	. . 3 ⊢ ((Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅) → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
21	20	3ad2ant3 1084	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
22		eqid 2622	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
23		eqid 2622	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
24		eqid 2622	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
25	22, 23, 2, 24, 3	issubgr 26163	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
26	25	3ad2ant1 1082	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
27	1, 16, 21, 26	mpbir3and 1245	1 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158   class class class wbr 4653  dom cdm 5114   ↾ cres 5116  Fun wfun 5882  ‘cfv 5888  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   SubGraph csubgr 26159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-edg 25940  df-subgr 26160

This theorem is referenced by:  0uhgrsubgr  26171