Theorem 0grsubgr 40502

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 3924	. . 3 ⊢ ∅ ⊆ (Vtx‘𝐺)
2		dm0 5260	. . . . 5 ⊢ dom ∅ = ∅
3	2	reseq2i 5314	. . . 4 ⊢ ((iEdg‘𝐺) ↾ dom ∅) = ((iEdg‘𝐺) ↾ ∅)
4		res0 5321	. . . 4 ⊢ ((iEdg‘𝐺) ↾ ∅) = ∅
5	3, 4	eqtr2i 2633	. . 3 ⊢ ∅ = ((iEdg‘𝐺) ↾ dom ∅)
6		0ss 3924	. . 3 ⊢ ∅ ⊆ 𝒫 ∅
7	1, 5, 6	3pm3.2i 1232	. 2 ⊢ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)
8		0ex 4718	. . 3 ⊢ ∅ ∈ V
9		vtxval0 25714	. . . . 5 ⊢ (Vtx‘∅) = ∅
10	9	eqcomi 2619	. . . 4 ⊢ ∅ = (Vtx‘∅)
11		eqid 2610	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
12		iedgval0 25715	. . . . 5 ⊢ (iEdg‘∅) = ∅
13	12	eqcomi 2619	. . . 4 ⊢ ∅ = (iEdg‘∅)
14		eqid 2610	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
15		edgaval 25794	. . . . . 6 ⊢ (∅ ∈ V → (Edg‘∅) = ran (iEdg‘∅))
16	8, 15	ax-mp 5	. . . . 5 ⊢ (Edg‘∅) = ran (iEdg‘∅)
17	12	rneqi 5273	. . . . 5 ⊢ ran (iEdg‘∅) = ran ∅
18		rn0 5298	. . . . 5 ⊢ ran ∅ = ∅
19	16, 17, 18	3eqtrri 2637	. . . 4 ⊢ ∅ = (Edg‘∅)
20	10, 11, 13, 14, 19	issubgr 40495	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ ∅ ∈ V) → (∅ SubGraph 𝐺 ↔ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)))
21	8, 20	mpan2 703	. 2 ⊢ (𝐺 ∈ 𝑊 → (∅ SubGraph 𝐺 ↔ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)))
22	7, 21	mpbiri 247	1 ⊢ (𝐺 ∈ 𝑊 → ∅ SubGraph 𝐺)

Syntax hints:   → wi 4   ↔ wb 195   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108   class class class wbr 4583  dom cdm 5038  ran crn 5039   ↾ cres 5040  ‘cfv 5804  Vtxcvtx 25673  iEdgciedg 25674  Edgcedga 25792   SubGraph csubgr 40491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-slot 15699  df-base 15700  df-edgf 25668  df-vtx 25675  df-iedg 25676  df-edga 25793  df-subgr 40492

This theorem is referenced by: (None)