Theorem uhgrspansubgr 16723

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 5089	. . 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 16722	. 2 |- ( ph -> (Edg ` S ) C_ ~P (Vtx ` S ) )
12	8	uhgrfun 16523	. . . 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 16704	. . 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 )

This proof depends on syntax axioms:   -> wi 4   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   ~Pcpw 3688   class class class wbr 4131   |` cres 4778   Fun wfun 5374   ` cfv 5380  Vtxcvtx 16458  iEdgciedg 16459  Edgcedg 16503  UHGraphcuhgr 16513   SubGraph csubgr 16699

This proof depends on 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 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296

This proof 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 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fo 5386  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-sub 8506  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-dec 9789  df-ndx 13413  df-slot 13414  df-base 13416  df-edgf 16451  df-vtx 16460  df-iedg 16461  df-edg 16504  df-uhgrm 16515  df-subgr 16700

This theorem is used by:  uhgrspan  16724  upgrspan  16725  umgrspan  16726  usgrspan  16727