Theorem uhgrspansubgr 16432

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 3268	. . 3 |- (Vtx ` S ) C_ (Vtx ` S )
2		uhgrspan.q	. . 3 |- ( ph -> (Vtx ` S ) = V )
3	1, 2	sseqtrid 3298	. 2 |- ( ph -> (Vtx ` S ) C_ V )
4		uhgrspan.r	. . 3 |- ( ph -> (iEdg ` S ) = ( E |` A ) )
5		resss 5082	. . 3 |- ( E |` A ) C_ E
6	4, 5	eqsstrdi 3300	. 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 16431	. 2 |- ( ph -> (Edg ` S ) C_ ~P (Vtx ` S ) )
12	8	uhgrfun 16232	. . . 4 |- ( G e. UHGraph -> Fun E )
13	10, 12	syl 14	. . 3 |- ( ph -> Fun E )
14		eqid 2238	. . . 4 |- (Vtx ` S ) = (Vtx ` S )
15		eqid 2238	. . . 4 |- (iEdg ` S ) = (iEdg ` S )
16		eqid 2238	. . . 4 |- (Edg ` S ) = (Edg ` S )
17	14, 7, 15, 8, 16	issubgr2 16413	. . 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 1278	. 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 1211	1 |- ( ph -> S SubGraph G )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   ~Pcpw 3685   class class class wbr 4125   |` cres 4771   Fun wfun 5366   ` cfv 5372  Vtxcvtx 16167  iEdgciedg 16168  Edgcedg 16212  UHGraphcuhgr 16222   SubGraph csubgr 16408

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-edg 16213  df-uhgrm 16224  df-subgr 16409

This theorem is referenced by:  uhgrspan  16433  upgrspan  16434  umgrspan  16435  usgrspan  16436