Theorem 0uhgrsubgr 29893

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 1166	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph))
2		0ss 4353	. . . 4 ⊢ ∅ ⊆ (Vtx‘𝐺)
3		sseq1 3938	. . . 4 ⊢ ((Vtx‘𝑆) = ∅ → ((Vtx‘𝑆) ⊆ (Vtx‘𝐺) ↔ ∅ ⊆ (Vtx‘𝐺)))
4	2, 3	mpbiri 261	. . 3 ⊢ ((Vtx‘𝑆) = ∅ → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
5	4	3ad2ant3 1153	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Vtx‘𝑆) ⊆ (Vtx‘𝐺))
6		eqid 2763	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
7	6	uhgrfun 29677	. . 3 ⊢ (𝑆 ∈ UHGraph → Fun (iEdg‘𝑆))
8	7	3ad2ant2 1152	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → Fun (iEdg‘𝑆))
9		edgval 29660	. . 3 ⊢ (Edg‘𝑆) = ran (iEdg‘𝑆)
10		uhgr0vb 29683	. . . . . . . 8 ⊢ ((𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝑆 ∈ UHGraph ↔ (iEdg‘𝑆) = ∅))
11		rneq 5926	. . . . . . . . 9 ⊢ ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ran ∅)
12		rn0 5916	. . . . . . . . 9 ⊢ ran ∅ = ∅
13	11, 12	eqtrdi 2814	. . . . . . . 8 ⊢ ((iEdg‘𝑆) = ∅ → ran (iEdg‘𝑆) = ∅)
14	10, 13	biimtrdi 256	. . . . . . 7 ⊢ ((𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (𝑆 ∈ UHGraph → ran (iEdg‘𝑆) = ∅))
15	14	ex 418	. . . . . 6 ⊢ (𝑆 ∈ UHGraph → ((Vtx‘𝑆) = ∅ → (𝑆 ∈ UHGraph → ran (iEdg‘𝑆) = ∅)))
16	15	pm2.43a 55	. . . . 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 1128	. . 3 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → ran (iEdg‘𝑆) = ∅)
19	9, 18	eqtrid 2810	. 2 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → (Edg‘𝑆) = ∅)
20		egrsubgr 29891	. 2 ⊢ (((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph) ∧ (Vtx‘𝑆) ⊆ (Vtx‘𝐺) ∧ (Fun (iEdg‘𝑆) ∧ (Edg‘𝑆) = ∅)) → 𝑆 SubGraph 𝐺)
21	1, 5, 8, 19, 20	syl112anc 1402	1 ⊢ ((𝐺 ∈ 𝑊 ∧ 𝑆 ∈ UHGraph ∧ (Vtx‘𝑆) = ∅) → 𝑆 SubGraph 𝐺)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ⊆ wss 3900  ∅c0 4281   class class class wbr 5106  ran crn 5657  Fun wfun 6555  ‘cfv 6561  Vtxcvtx 29607  iEdgciedg 29608  Edgcedg 29658  UHGraphcuhgr 29667   SubGraph csubgr 29881

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-edg 29659  df-uhgr 29669  df-subgr 29882

This theorem is used by: (None)