Theorem egrsubgr 29536

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 1153	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
2		eqid 2765	. . . . . . 7 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
3		eqid 2765	. . . . . . 7 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
4	2, 3	edg0iedg0 29314	. . . . . 6 ⊢ (Fun (iEdg‘𝑆) → ((Edg‘𝑆) = ∅ ↔ (iEdg‘𝑆) = ∅))
5	4	adantl 486	. . . . 5 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ Fun (iEdg‘𝑆)) → ((Edg‘𝑆) = ∅ ↔ (iEdg‘𝑆) = ∅))
6		res0 5973	. . . . . . 7 ⊢ ((iEdg‘𝐺) ↾ ∅) = ∅
7	6	eqcomi 2774	. . . . . 6 ⊢ ∅ = ((iEdg‘𝐺) ↾ ∅)
8		id 23	. . . . . 6 ⊢ ((iEdg‘𝑆) = ∅ → (iEdg‘𝑆) = ∅)
9		dmeq 5884	. . . . . . . 8 ⊢ ((iEdg‘𝑆) = ∅ → dom (iEdg‘𝑆) = dom ∅)
10		dm0 5901	. . . . . . . 8 ⊢ dom ∅ = ∅
11	9, 10	eqtrdi 2816	. . . . . . 7 ⊢ ((iEdg‘𝑆) = ∅ → dom (iEdg‘𝑆) = ∅)
12	11	reseq2d 5969	. . . . . 6 ⊢ ((iEdg‘𝑆) = ∅ → ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)) = ((iEdg‘𝐺) ↾ ∅))
13	7, 8, 12	3eqtr4a 2826	. . . . 5 ⊢ ((iEdg‘𝑆) = ∅ → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
14	5, 13	biimtrdi 256	. . . 4 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ Fun (iEdg‘𝑆)) → ((Edg‘𝑆) = ∅ → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆))))
15	14	impr 459	. . 3 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
16	15	3adant2 1147	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (iEdg‘𝑆) = ((iEdg‘𝐺) ↾ dom (iEdg‘𝑆)))
17		0ss 4357	. . . . 5 ⊢ ∅ ⊆ 𝒫 (Vtx‘𝑆)
18		sseq1 3964	. . . . 5 ⊢ ((Edg‘𝑆) = ∅ → ((Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆) ↔ ∅ ⊆ 𝒫 (Vtx‘𝑆)))
19	17, 18	mpbiri 261	. . . 4 ⊢ ((Edg‘𝑆) = ∅ → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
20	19	adantl 486	. . 3 ⊢ ((Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅) → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
21	20	3ad2ant3 1151	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
22		eqid 2765	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
23		eqid 2765	. . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
24		eqid 2765	. . . 4 ⊢ (iEdg‘𝐺) = (iEdg‘𝐺)
25	22, 23, 2, 24, 3	issubgr 29530	. . 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 1149	. 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 1359	1 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ 𝑈) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ⊆ wss 3907  ∅c0 4288  𝒫 cpw 4558   class class class wbr 5105  dom cdm 5652   ↾ cres 5654  Fun wfun 6519  ‘cfv 6525  Vtxcvtx 29255  iEdgciedg 29256  Edgcedg 29306   SubGraph csubgr 29526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-iota 6481  df-fun 6527  df-fv 6533  df-edg 29307  df-subgr 29527

This theorem is referenced by:  0uhgrsubgr  29538