Theorem 0grsubgr 26170

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 3972	. . 3 ⊢ ∅ ⊆ (Vtx‘𝐺)
2		dm0 5339	. . . . 5 ⊢ dom ∅ = ∅
3	2	reseq2i 5393	. . . 4 ⊢ ((iEdg‘𝐺) ↾ dom ∅) = ((iEdg‘𝐺) ↾ ∅)
4		res0 5400	. . . 4 ⊢ ((iEdg‘𝐺) ↾ ∅) = ∅
5	3, 4	eqtr2i 2645	. . 3 ⊢ ∅ = ((iEdg‘𝐺) ↾ dom ∅)
6		0ss 3972	. . 3 ⊢ ∅ ⊆ 𝒫 ∅
7	1, 5, 6	3pm3.2i 1239	. 2 ⊢ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)
8		0ex 4790	. . 3 ⊢ ∅ ∈ V
9		vtxval0 25931	. . . . 5 ⊢ (Vtx‘∅) = ∅
10	9	eqcomi 2631	. . . 4 ⊢ ∅ = (Vtx‘∅)
11		eqid 2622	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
12		iedgval0 25932	. . . . 5 ⊢ (iEdg‘∅) = ∅
13	12	eqcomi 2631	. . . 4 ⊢ ∅ = (iEdg‘∅)
14		eqid 2622	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
15		edgval 25941	. . . . 5 ⊢ (Edg‘∅) = ran (iEdg‘∅)
16	12	rneqi 5352	. . . . 5 ⊢ ran (iEdg‘∅) = ran ∅
17		rn0 5377	. . . . 5 ⊢ ran ∅ = ∅
18	15, 16, 17	3eqtrri 2649	. . . 4 ⊢ ∅ = (Edg‘∅)
19	10, 11, 13, 14, 18	issubgr 26163	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ ∅ ∈ V) → (∅ SubGraph 𝐺 ↔ (∅ ⊆ (Vtx‘𝐺) ∧ ∅ = ((iEdg‘𝐺) ↾ dom ∅) ∧ ∅ ⊆ 𝒫 ∅)))
20	8, 19	mpan2 707	. 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 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  𝒫 cpw 4158   class class class wbr 4653  dom cdm 5114  ran crn 5115   ↾ cres 5116  ‘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-ne 2795  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-slot 15861  df-base 15863  df-edgf 25868  df-vtx 25876  df-iedg 25877  df-edg 25940  df-subgr 26160

This theorem is referenced by: (None)