Theorem egrsubgr 28534

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 1138	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
2		eqid 2733	. . . . . . 7 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
3		eqid 2733	. . . . . . 7 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
4	2, 3	edg0iedg0 28315	. . . . . 6 ⊢ (Fun (iEdg‘𝑆) → ((Edg‘𝑆) = ∅ ↔ (iEdg‘𝑆) = ∅))
5	4	adantl 483	. . . . 5 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ Fun (iEdg‘𝑆)) → ((Edg‘𝑆) = ∅ ↔ (iEdg‘𝑆) = ∅))
6		res0 5986	. . . . . . 7 ⊢ ((iEdg‘𝐺) ↾ ∅) = ∅
7	6	eqcomi 2742	. . . . . 6 ⊢ ∅ = ((iEdg‘𝐺) ↾ ∅)
8		id 22	. . . . . 6 ⊢ ((iEdg‘𝑆) = ∅ → (iEdg‘𝑆) = ∅)
9		dmeq 5904	. . . . . . . 8 ⊢ ((iEdg‘𝑆) = ∅ → dom (iEdg‘𝑆) = dom ∅)
10		dm0 5921	. . . . . . . 8 ⊢ dom ∅ = ∅
11	9, 10	eqtrdi 2789	. . . . . . 7 ⊢ ((iEdg‘𝑆) = ∅ → dom (iEdg‘𝑆) = ∅)
12	11	reseq2d 5982	. . . . . 6 ⊢ ((iEdg‘𝑆) = ∅ → ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) = ((iEdg‘𝐺) ↾ ∅))
13	7, 8, 12	3eqtr4a 2799	. . . . 5 ⊢ ((iEdg‘𝑆) = ∅ → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
14	5, 13	syl6bi 253	. . . 4 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ Fun (iEdg‘𝑆)) → ((Edg‘𝑆) = ∅ → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))))
15	14	impr 456	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
16	15	3adant2 1132	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
17		0ss 4397	. . . . 5 ⊢ ∅ ⊆ 𝒫 (Vtx‘𝑆)
18		sseq1 4008	. . . . 5 ⊢ ((Edg‘𝑆) = ∅ → ((Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆) ↔ ∅ ⊆ 𝒫 (Vtx‘𝑆)))
19	17, 18	mpbiri 258	. . . 4 ⊢ ((Edg‘𝑆) = ∅ → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
20	19	adantl 483	. . 3 ⊢ ((Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅) → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
21	20	3ad2ant3 1136	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
22		eqid 2733	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
23		eqid 2733	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
24		eqid 2733	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
25	22, 23, 2, 24, 3	issubgr 28528	. . 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 1134	. 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 1343	1 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   ⊆ wss 3949  ∅c0 4323  𝒫 cpw 4603   class class class wbr 5149  dom cdm 5677   ↾ cres 5679  Fun wfun 6538  ‘cfv 6544  Vtxcvtx 28256  iEdgciedg 28257  Edgcedg 28307   SubGraph csubgr 28524

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-iota 6496  df-fun 6546  df-fv 6552  df-edg 28308  df-subgr 28525

This theorem is referenced by:  0uhgrsubgr  28536