Theorem uhgrspansubgr 16134

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 3247	. . 3 |- (Vtx ` S ) C_ (Vtx ` S )
2		uhgrspan.q	. . 3 |- ( ph -> (Vtx ` S ) = V )
3	1, 2	sseqtrid 3277	. 2 |- ( ph -> (Vtx ` S ) C_ V )
4		uhgrspan.r	. . 3 |- ( ph -> (iEdg ` S ) = ( E |` A ) )
5		resss 5037	. . 3 |- ( E |` A ) C_ E
6	4, 5	eqsstrdi 3279	. 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 16133	. 2 |- ( ph -> (Edg ` S ) C_ ~P (Vtx ` S ) )
12	8	uhgrfun 15934	. . . 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 16115	. . 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 1273	. 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 1206	1 |- ( ph -> S SubGraph G )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   C_ wss 3200   ~Pcpw 3652   class class class wbr 4088   |` cres 4727   Fun wfun 5320   ` cfv 5326  Vtxcvtx 15869  iEdgciedg 15870  Edgcedg 15914  UHGraphcuhgr 15924   SubGraph csubgr 16110

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-sub 8352  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-dec 9612  df-ndx 13090  df-slot 13091  df-base 13093  df-edgf 15862  df-vtx 15871  df-iedg 15872  df-edg 15915  df-uhgrm 15926  df-subgr 16111

This theorem is referenced by:  uhgrspan  16135  upgrspan  16136  umgrspan  16137  usgrspan  16138