Theorem 0grsubgr 26361

Description: The null graph (represented by an empty set) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.)

Assertion

Ref	Expression
0grsubgr	⊢ (𝐺 ∈ 𝑊 → ∅ SubGraph 𝐺)

Proof of Theorem 0grsubgr

Step	Hyp	Ref	Expression
1		0ss 4107	. . 3 ⊢ ∅ ⊆ (Vtx‘𝐺)
2		dm0 5486	. . . . 5 ⊢ dom ∅ = ∅
3	2	reseq2i 5540	. . . 4 ⊢ ((iEdg‘𝐺) ↾ dom ∅) = ((iEdg‘𝐺) ↾ ∅)
4		res0 5547	. . . 4 ⊢ ((iEdg‘𝐺) ↾ ∅) = ∅
5	3, 4	eqtr2i 2775	. . 3 ⊢ ∅ = ((iEdg‘𝐺) ↾ dom ∅)
6		0ss 4107	. . 3 ⊢ ∅ ⊆ 𝒫 ∅
7	1, 5, 6	3pm3.2i 1421	. 2 ⊢ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)
8		0ex 4934	. . 3 ⊢ ∅ ∈ V
9		vtxval0 26122	. . . . 5 ⊢ (Vtx‘∅) = ∅
10	9	eqcomi 2761	. . . 4 ⊢ ∅ = (Vtx‘∅)
11		eqid 2752	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
12		iedgval0 26123	. . . . 5 ⊢ (iEdg‘∅) = ∅
13	12	eqcomi 2761	. . . 4 ⊢ ∅ = (iEdg‘∅)
14		eqid 2752	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
15		edgval 26132	. . . . 5 ⊢ (Edg‘∅) = ran (iEdg‘∅)
16	12	rneqi 5499	. . . . 5 ⊢ ran (iEdg‘∅) = ran ∅
17		rn0 5524	. . . . 5 ⊢ ran ∅ = ∅
18	15, 16, 17	3eqtrri 2779	. . . 4 ⊢ ∅ = (Edg‘∅)
19	10, 11, 13, 14, 18	issubgr 26354	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ ∅ ∈ V) → (∅ SubGraph 𝐺 ↔ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)))
20	8, 19	mpan2 709	. 2 ⊢ (𝐺 ∈ 𝑊 → (∅ SubGraph 𝐺 ↔ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)))
21	7, 20	mpbiri 248	1 ⊢ (𝐺 ∈ 𝑊 → ∅ SubGraph 𝐺)

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1072   = wceq 1624   ∈ wcel 2131  Vcvv 3332   ⊆ wss 3707  ∅c0 4050  𝒫 cpw 4294   class class class wbr 4796  dom cdm 5258  ran crn 5259   ↾ cres 5260  ‘cfv 6041  Vtxcvtx 26065  iEdgciedg 26066  Edgcedg 26130   SubGraph csubgr 26350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047  ax-un 7106

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-csb 3667  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-rn 5269  df-res 5270  df-iota 6004  df-fun 6043  df-fv 6049  df-slot 16055  df-base 16057  df-edgf 26059  df-vtx 26067  df-iedg 26068  df-edg 26131  df-subgr 26351

This theorem is referenced by: (None)