Theorem 0uhgrsubgr 29296

Description: The null graph (as hypergraph) is a subgraph of all graphs. (Contributed by AV, 17-Nov-2020.) (Proof shortened by AV, 28-Nov-2020.)

Assertion

Ref	Expression
0uhgrsubgr	⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → 𝑆 SubGraph 𝐺)

Proof of Theorem 0uhgrsubgr

Step	Hyp	Ref	Expression
1		3simpa 1149	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph))
2		0ss 4400	. . . 4 ⊢ ∅ ⊆ (Vtx‘𝐺)
3		sseq1 4009	. . . 4 ⊢ ((Vtx‘𝑆) = ∅ → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ↔ ∅ ⊆ (Vtx‘𝐺)))
4	2, 3	mpbiri 258	. . 3 ⊢ ((Vtx‘𝑆) = ∅ → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
5	4	3ad2ant3 1136	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
6		eqid 2737	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
7	6	uhgrfun 29083	. . 3 ⊢ (𝑆 ∈ UHGraph → Fun (iEdg‘𝑆))
8	7	3ad2ant2 1135	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → Fun (iEdg‘𝑆))
9		edgval 29066	. . 3 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
10		uhgr0vb 29089	. . . . . . . 8 ⊢ ((𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆) = ∅))
11		rneq 5947	. . . . . . . . 9 ⊢ ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ran ∅)
12		rn0 5936	. . . . . . . . 9 ⊢ ran ∅ = ∅
13	11, 12	eqtrdi 2793	. . . . . . . 8 ⊢ ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ∅)
14	10, 13	biimtrdi 253	. . . . . . 7 ⊢ ((𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝑆 ∈ UHGraph → ran (iEdg‘𝑆) = ∅))
15	14	ex 412	. . . . . 6 ⊢ (𝑆 ∈ UHGraph → ((Vtx‘𝑆) = ∅ → (𝑆 ∈ UHGraph → ran (iEdg‘𝑆) = ∅)))
16	15	pm2.43a 54	. . . . 5 ⊢ (𝑆 ∈ UHGraph → ((Vtx‘𝑆) = ∅ → ran (iEdg‘𝑆) = ∅))
17	16	a1i 11	. . . 4 ⊢ (𝐺 ∈ 𝑊 → (𝑆 ∈ UHGraph → ((Vtx‘𝑆) = ∅ → ran (iEdg‘𝑆) = ∅)))
18	17	3imp 1111	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → ran (iEdg‘𝑆) = ∅)
19	9, 18	eqtrid 2789	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Edg‘𝑆) = ∅)
20		egrsubgr 29294	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺)
21	1, 5, 8, 19, 20	syl112anc 1376	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → 𝑆 SubGraph 𝐺)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540   ∈ wcel 2108   ⊆ wss 3951  ∅c0 4333   class class class wbr 5143  ran crn 5686  Fun wfun 6555  ‘cfv 6561  Vtxcvtx 29013  iEdgciedg 29014  Edgcedg 29064  UHGraphcuhgr 29073   SubGraph csubgr 29284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-edg 29065  df-uhgr 29075  df-subgr 29285

This theorem is referenced by: (None)