Theorem uhgrspansubgr 29269

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 3952	. . 3 ⊢ (Vtx‘𝑆) ⊆ (Vtx‘𝑆)
2		uhgrspan.q	. . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
3	1, 2	sseqtrid 3972	. 2 ⊢ (𝜑 → (Vtx‘𝑆) ⊆ 𝑉)
4		uhgrspan.r	. . 3 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴))
5		resss 5949	. . 3 ⊢ (𝐸 ↾ 𝐴) ⊆ 𝐸
6	4, 5	eqsstrdi 3974	. 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 29268	. 2 ⊢ (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
12	8	uhgrfun 29044	. . . 4 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
13	10, 12	syl 17	. . 3 ⊢ (𝜑 → Fun 𝐸)
14		eqid 2731	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
15		eqid 2731	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
16		eqid 2731	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
17	14, 7, 15, 8, 16	issubgr2 29250	. . 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 1373	. 2 ⊢ (𝜑 → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
19	3, 6, 11, 18	mpbir3and 1343	1 ⊢ (𝜑 → 𝑆 SubGraph 𝐺)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ⊆ wss 3897  𝒫 cpw 4547   class class class wbr 5089   ↾ cres 5616  Fun wfun 6475  ‘cfv 6481  Vtxcvtx 28974  iEdgciedg 28975  Edgcedg 29025  UHGraphcuhgr 29034   SubGraph csubgr 29245

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-edg 29026  df-uhgr 29036  df-subgr 29246

This theorem is referenced by:  uhgrspan  29270  upgrspan  29271  umgrspan  29272  usgrspan  29273