Theorem 0grsubgr 26575

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 4197	. . 3 ⊢ ∅ ⊆ (Vtx‘𝐺)
2		dm0 5571	. . . . 5 ⊢ dom ∅ = ∅
3	2	reseq2i 5626	. . . 4 ⊢ ((iEdg‘𝐺) ↾ dom ∅) = ((iEdg‘𝐺) ↾ ∅)
4		res0 5633	. . . 4 ⊢ ((iEdg‘𝐺) ↾ ∅) = ∅
5	3, 4	eqtr2i 2850	. . 3 ⊢ ∅ = ((iEdg‘𝐺) ↾ dom ∅)
6		0ss 4197	. . 3 ⊢ ∅ ⊆ 𝒫 ∅
7	1, 5, 6	3pm3.2i 1442	. 2 ⊢ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)
8		0ex 5014	. . 3 ⊢ ∅ ∈ V
9		vtxval0 26337	. . . . 5 ⊢ (Vtx‘∅) = ∅
10	9	eqcomi 2834	. . . 4 ⊢ ∅ = (Vtx‘∅)
11		eqid 2825	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
12		iedgval0 26338	. . . . 5 ⊢ (iEdg‘∅) = ∅
13	12	eqcomi 2834	. . . 4 ⊢ ∅ = (iEdg‘∅)
14		eqid 2825	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
15		edgval 26347	. . . . 5 ⊢ (Edg‘∅) = ran (iEdg‘∅)
16	12	rneqi 5584	. . . . 5 ⊢ ran (iEdg‘∅) = ran ∅
17		rn0 5610	. . . . 5 ⊢ ran ∅ = ∅
18	15, 16, 17	3eqtrri 2854	. . . 4 ⊢ ∅ = (Edg‘∅)
19	10, 11, 13, 14, 18	issubgr 26568	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ ∅ ∈ V) → (∅ SubGraph 𝐺 ↔ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)))
20	8, 19	mpan2 682	. 2 ⊢ (𝐺 ∈ 𝑊 → (∅ SubGraph 𝐺 ↔ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)))
21	7, 20	mpbiri 250	1 ⊢ (𝐺 ∈ 𝑊 → ∅ SubGraph 𝐺)

Syntax hints:   → wi 4   ↔ wb 198   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164  Vcvv 3414   ⊆ wss 3798  ∅c0 4144  𝒫 cpw 4378   class class class wbr 4873  dom cdm 5342  ran crn 5343   ↾ cres 5344  ‘cfv 6123  Vtxcvtx 26294  iEdgciedg 26295  Edgcedg 26345   SubGraph csubgr 26564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-iota 6086  df-fun 6125  df-fv 6131  df-slot 16226  df-base 16228  df-edgf 26288  df-vtx 26296  df-iedg 26297  df-edg 26346  df-subgr 26565

This theorem is referenced by: (None)