Theorem uhgrspansubgr 26595

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 3848	. . 3 ⊢ (Vtx‘𝑆) ⊆ (Vtx‘𝑆)
2		uhgrspan.q	. . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
3	1, 2	syl5sseq 3878	. 2 ⊢ (𝜑 → (Vtx‘𝑆) ⊆ 𝑉)
4		uhgrspan.r	. . 3 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴))
5		resss 5662	. . 3 ⊢ (𝐸 ↾ 𝐴) ⊆ 𝐸
6	4, 5	syl6eqss 3880	. 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 26594	. 2 ⊢ (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
12	8	uhgrfun 26371	. . . 4 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
13	10, 12	syl 17	. . 3 ⊢ (𝜑 → Fun 𝐸)
14		eqid 2825	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
15		eqid 2825	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
16		eqid 2825	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
17	14, 7, 15, 8, 16	issubgr2 26576	. . 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 1494	. 2 ⊢ (𝜑 → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
19	3, 6, 11, 18	mpbir3and 1446	1 ⊢ (𝜑 → 𝑆 SubGraph 𝐺)

Syntax hints:   → wi 4   ↔ wb 198   ∧ w3a 1111   = wceq 1656   ∈ wcel 2164   ⊆ wss 3798  𝒫 cpw 4380   class class class wbr 4875   ↾ cres 5348  Fun wfun 6121  ‘cfv 6127  Vtxcvtx 26301  iEdgciedg 26302  Edgcedg 26352  UHGraphcuhgr 26361   SubGraph csubgr 26571

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-fv 6135  df-edg 26353  df-uhgr 26363  df-subgr 26572

This theorem is referenced by:  uhgrspan  26596  upgrspan  26597  umgrspan  26598  usgrspan  26599