Theorem uhgrspansubgr 16201

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	|- ( ph -> S e. W )
uhgrspan.q	|- ( ph -> (Vtx ` S ) = V )
uhgrspan.r	|- ( ph -> (iEdg ` S ) = ( E |` A ) )
uhgrspan.g	|- ( ph -> G e. UHGraph )

Assertion

Ref	Expression
uhgrspansubgr	|- ( ph -> S SubGraph G )

Proof of Theorem uhgrspansubgr

Step	Hyp	Ref	Expression
1		ssid 3248	. . 3 |- (Vtx ` S ) C_ (Vtx ` S )
2		uhgrspan.q	. . 3 |- ( ph -> (Vtx ` S ) = V )
3	1, 2	sseqtrid 3278	. 2 |- ( ph -> (Vtx ` S ) C_ V )
4		uhgrspan.r	. . 3 |- ( ph -> (iEdg ` S ) = ( E |` A ) )
5		resss 5043	. . 3 |- ( E |` A ) C_ E
6	4, 5	eqsstrdi 3280	. 2 |- ( ph -> (iEdg ` S ) C_ E )
7		uhgrspan.v	. . 3 |- V = (Vtx ` G )
8		uhgrspan.e	. . 3 |- E = (iEdg ` G )
9		uhgrspan.s	. . 3 |- ( ph -> S e. W )
10		uhgrspan.g	. . 3 |- ( ph -> G e. UHGraph )
11	7, 8, 9, 2, 4, 10	uhgrspansubgrlem 16200	. 2 |- ( ph -> (Edg ` S ) C_ ~P (Vtx ` S ) )
12	8	uhgrfun 16001	. . . 4 |- ( G e. UHGraph -> Fun E )
13	10, 12	syl 14	. . 3 |- ( ph -> Fun E )
14		eqid 2231	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
15		eqid 2231	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
16		eqid 2231	. . . 4 |- (Edg ` S ) = (Edg ` S )
17	14, 7, 15, 8, 16	issubgr2 16182	. . 3 |- ( ( G e. UHGraph /\ Fun E /\ S e. W ) -> ( S SubGraph G <-> ( (Vtx ` S ) C_ V /\ (iEdg ` S ) C_ E /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) ) )
18	10, 13, 9, 17	syl3anc 1274	. 2 |- ( ph -> ( S SubGraph G <-> ( (Vtx ` S ) C_ V /\ (iEdg ` S ) C_ E /\ (Edg ` S ) C_ ~P (Vtx ` S ) ) ) )
19	3, 6, 11, 18	mpbir3and 1207	1 |- ( ph -> S SubGraph G )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   C_ wss 3201   ~Pcpw 3656   class class class wbr 4093   |` cres 4733   Fun wfun 5327   ` cfv 5333  Vtxcvtx 15936  iEdgciedg 15937  Edgcedg 15981  UHGraphcuhgr 15991   SubGraph csubgr 16177

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-sub 8394  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-dec 9656  df-ndx 13148  df-slot 13149  df-base 13151  df-edgf 15929  df-vtx 15938  df-iedg 15939  df-edg 15982  df-uhgrm 15993  df-subgr 16178

This theorem is referenced by:  uhgrspan  16202  upgrspan  16203  umgrspan  16204  usgrspan  16205