Theorem uhgrspansubgr 29380

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 3958	. . 3 ⊢ (Vtx‘𝑆) ⊆ (Vtx‘𝑆)
2		uhgrspan.q	. . 3 ⊢ (𝜑 → (Vtx‘𝑆) = 𝑉)
3	1, 2	sseqtrid 3978	. 2 ⊢ (𝜑 → (Vtx‘𝑆) ⊆ 𝑉)
4		uhgrspan.r	. . 3 ⊢ (𝜑 → (iEdg‘𝑆) = (𝐸 ↾ 𝐴))
5		resss 5968	. . 3 ⊢ (𝐸 ↾ 𝐴) ⊆ 𝐸
6	4, 5	eqsstrdi 3980	. 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 29379	. 2 ⊢ (𝜑 → (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))
12	8	uhgrfun 29155	. . . 4 ⊢ (𝐺 ∈ UHGraph → Fun 𝐸)
13	10, 12	syl 17	. . 3 ⊢ (𝜑 → Fun 𝐸)
14		eqid 2737	. . . 4 ⊢ (Vtx‘𝑆) = (Vtx‘𝑆)
15		eqid 2737	. . . 4 ⊢ (iEdg‘𝑆) = (iEdg‘𝑆)
16		eqid 2737	. . . 4 ⊢ (Edg‘𝑆) = (Edg‘𝑆)
17	14, 7, 15, 8, 16	issubgr2 29361	. . 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 1374	. 2 ⊢ (𝜑 → (𝑆 SubGraph 𝐺 ↔ ((Vtx‘𝑆) ⊆ 𝑉 ∧ (iEdg‘𝑆) ⊆ 𝐸 ∧ (Edg‘𝑆) ⊆ 𝒫 (Vtx‘𝑆))))
19	3, 6, 11, 18	mpbir3and 1344	1 ⊢ (𝜑 → 𝑆 SubGraph 𝐺)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114   ⊆ wss 3903  𝒫 cpw 4556   class class class wbr 5100   ↾ cres 5634  Fun wfun 6494  ‘cfv 6500  Vtxcvtx 29085  iEdgciedg 29086  Edgcedg 29136  UHGraphcuhgr 29145   SubGraph csubgr 29356

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-edg 29137  df-uhgr 29147  df-subgr 29357

This theorem is referenced by:  uhgrspan  29381  upgrspan  29382  umgrspan  29383  usgrspan  29384