Theorem uhgrspansubgr 27076

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 3992	. . 3 ⊢ (Vtx‘𝑆) ⊆ (Vtx‘𝑆)
2		uhgrspan.q	. . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
3	1, 2	sseqtrid 4022	. 2 ⊢ (𝜑 → (Vtx‘𝑆) ⊆ 𝑉)
4		uhgrspan.r	. . 3 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴))
5		resss 5881	. . 3 ⊢ (𝐸 ↾ 𝐴) ⊆ 𝐸
6	4, 5	eqsstrdi 4024	. 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 27075	. 2 ⊢ (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
12	8	uhgrfun 26854	. . . 4 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
13	10, 12	syl 17	. . 3 ⊢ (𝜑 → Fun 𝐸)
14		eqid 2824	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
15		eqid 2824	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
16		eqid 2824	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
17	14, 7, 15, 8, 16	issubgr2 27057	. . 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 1367	. 2 ⊢ (𝜑 → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
19	3, 6, 11, 18	mpbir3and 1338	1 ⊢ (𝜑 → 𝑆 SubGraph 𝐺)

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1083   = wceq 1536   ∈ wcel 2113   ⊆ wss 3939  𝒫 cpw 4542   class class class wbr 5069   ↾ cres 5560  Fun wfun 6352  ‘cfv 6358  Vtxcvtx 26784  iEdgciedg 26785  Edgcedg 26835  UHGraphcuhgr 26844   SubGraph csubgr 27052

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 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2796  ax-sep 5206  ax-nul 5213  ax-pow 5269  ax-pr 5333  ax-un 7464

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-mo 2621  df-eu 2653  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2966  df-ral 3146  df-rex 3147  df-rab 3150  df-v 3499  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4471  df-pw 4544  df-sn 4571  df-pr 4573  df-op 4577  df-uni 4842  df-br 5070  df-opab 5132  df-mpt 5150  df-id 5463  df-xp 5564  df-rel 5565  df-cnv 5566  df-co 5567  df-dm 5568  df-rn 5569  df-res 5570  df-iota 6317  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-edg 26836  df-uhgr 26846  df-subgr 27053

This theorem is referenced by:  uhgrspan  27077  upgrspan  27078  umgrspan  27079  usgrspan  27080