Theorem uhgrspansubgr 29309

Description: A spanning subgraph 𝑆 of a hypergraph 𝐺 is actually a subgraph of 𝐺. A subgraph 𝑆 of a graph 𝐺 which has the same vertices as 𝐺 and is obtained by removing some edges of 𝐺 is called a spanning subgraph (see section I.1 in [Bollobas] p. 2 and section 1.1 in [Diestel] p. 4). Formally, the edges are "removed" by restricting the edge function of the original graph by an arbitrary class (which actually needs not to be a subset of the domain of the edge function). (Contributed by AV, 18-Nov-2020.) (Proof shortened by AV, 21-Nov-2020.)

Hypotheses

Ref	Expression
uhgrspan.v	⊢ 𝑉 = (Vtx‘𝐺)
uhgrspan.e	⊢ 𝐸 = (iEdg‘𝐺)
uhgrspan.s	⊢ (𝜑 → 𝑆 ∈ 𝑊)
uhgrspan.q	⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
uhgrspan.r	⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴))
uhgrspan.g	⊢ (𝜑 → 𝐺 ∈ UHGraph)

Assertion

Ref	Expression
uhgrspansubgr	⊢ (𝜑 → 𝑆 SubGraph 𝐺)

Proof of Theorem uhgrspansubgr

Step	Hyp	Ref	Expression
1		ssid 4005	. . 3 ⊢ (Vtx‘𝑆) ⊆ (Vtx‘𝑆)
2		uhgrspan.q	. . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
3	1, 2	sseqtrid 4025	. 2 ⊢ (𝜑 → (Vtx‘𝑆) ⊆ 𝑉)
4		uhgrspan.r	. . 3 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴))
5		resss 6018	. . 3 ⊢ (𝐸 ↾ 𝐴) ⊆ 𝐸
6	4, 5	eqsstrdi 4027	. 2 ⊢ (𝜑 → (iEdg‘𝑆) ⊆ 𝐸)
7		uhgrspan.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
8		uhgrspan.e	. . 3 ⊢ 𝐸 = (iEdg‘𝐺)
9		uhgrspan.s	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑊)
10		uhgrspan.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ UHGraph)
11	7, 8, 9, 2, 4, 10	uhgrspansubgrlem 29308	. 2 ⊢ (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
12	8	uhgrfun 29084	. . . 4 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
13	10, 12	syl 17	. . 3 ⊢ (𝜑 → Fun 𝐸)
14		eqid 2736	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
15		eqid 2736	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
16		eqid 2736	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
17	14, 7, 15, 8, 16	issubgr2 29290	. . 3 ⊢ ((𝐺 ∈ UHGraph ∧ Fun 𝐸 ∧ 𝑆 ∈ 𝑊) → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
18	10, 13, 9, 17	syl3anc 1372	. 2 ⊢ (𝜑 → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
19	3, 6, 11, 18	mpbir3and 1342	1 ⊢ (𝜑 → 𝑆 SubGraph 𝐺)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107   ⊆ wss 3950  𝒫 cpw 4599   class class class wbr 5142   ↾ cres 5686  Fun wfun 6554  ‘cfv 6560  Vtxcvtx 29014  iEdgciedg 29015  Edgcedg 29065  UHGraphcuhgr 29074   SubGraph csubgr 29285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-edg 29066  df-uhgr 29076  df-subgr 29286

This theorem is referenced by:  uhgrspan  29310  upgrspan  29311  umgrspan  29312  usgrspan  29313