Theorem uhgrissubgr 16273

Description: The property of a hypergraph to be a subgraph. (Contributed by AV, 19-Nov-2020.)

Hypotheses

Ref	Expression
uhgrissubgr.v	|- V = (Vtx ` S )
uhgrissubgr.a	|- A = (Vtx ` G )
uhgrissubgr.i	|- I = (iEdg ` S )
uhgrissubgr.b	|- B = (iEdg ` G )

Assertion

Ref	Expression
uhgrissubgr	|- ( ( G e. W /\ Fun B /\ S e. UHGraph ) -> ( S SubGraph G <-> ( V C_ A /\ I C_ B ) ) )

Proof of Theorem uhgrissubgr

Dummy variable e is distinct from all other variables.

Step	Hyp	Ref	Expression
1		uhgrissubgr.v	. . . 4 |- V = (Vtx ` S )
2		uhgrissubgr.a	. . . 4 |- A = (Vtx ` G )
3		uhgrissubgr.i	. . . 4 |- I = (iEdg ` S )
4		uhgrissubgr.b	. . . 4 |- B = (iEdg ` G )
5		eqid 2234	. . . 4 |- (Edg ` S ) = (Edg ` S )
6	1, 2, 3, 4, 5	subgrprop2 16272	. . 3 |- ( S SubGraph G -> ( V C_ A /\ I C_ B /\ (Edg ` S ) C_ ~P V ) )
7		3simpa 1021	. . 3 |- ( ( V C_ A /\ I C_ B /\ (Edg ` S ) C_ ~P V ) -> ( V C_ A /\ I C_ B ) )
8	6, 7	syl 14	. 2 |- ( S SubGraph G -> ( V C_ A /\ I C_ B ) )
9		simprl 531	. . . 4 |- ( ( ( G e. W /\ Fun B /\ S e. UHGraph ) /\ ( V C_ A /\ I C_ B ) ) -> V C_ A )
10		simp2 1025	. . . . . 6 |- ( ( G e. W /\ Fun B /\ S e. UHGraph ) -> Fun B )
11		simpr 110	. . . . . 6 |- ( ( V C_ A /\ I C_ B ) -> I C_ B )
12		funssres 5397	. . . . . 6 |- ( ( Fun B /\ I C_ B ) -> ( B |` dom I ) = I )
13	10, 11, 12	syl2an 289	. . . . 5 |- ( ( ( G e. W /\ Fun B /\ S e. UHGraph ) /\ ( V C_ A /\ I C_ B ) ) -> ( B |` dom I ) = I )
14	13	eqcomd 2240	. . . 4 |- ( ( ( G e. W /\ Fun B /\ S e. UHGraph ) /\ ( V C_ A /\ I C_ B ) ) -> I = ( B |` dom I ) )
15		edguhgr 16149	. . . . . . . . 9 |- ( ( S e. UHGraph /\ e e. (Edg ` S ) ) -> e e. ~P (Vtx ` S ) )
16	15	ex 115	. . . . . . . 8 |- ( S e. UHGraph -> ( e e. (Edg ` S ) -> e e. ~P (Vtx ` S ) ) )
17	1	pweqi 3675	. . . . . . . . 9 |- ~P V = ~P (Vtx ` S )
18	17	eleq2i 2301	. . . . . . . 8 |- ( e e. ~P V <-> e e. ~P (Vtx ` S ) )
19	16, 18	imbitrrdi 162	. . . . . . 7 |- ( S e. UHGraph -> ( e e. (Edg ` S ) -> e e. ~P V ) )
20	19	ssrdv 3246	. . . . . 6 |- ( S e. UHGraph -> (Edg ` S ) C_ ~P V )
21	20	3ad2ant3 1047	. . . . 5 |- ( ( G e. W /\ Fun B /\ S e. UHGraph ) -> (Edg ` S ) C_ ~P V )
22	21	adantr 276	. . . 4 |- ( ( ( G e. W /\ Fun B /\ S e. UHGraph ) /\ ( V C_ A /\ I C_ B ) ) -> (Edg ` S ) C_ ~P V )
23	1, 2, 3, 4, 5	issubgr 16269	. . . . . 6 |- ( ( G e. W /\ S e. UHGraph ) -> ( S SubGraph G <-> ( V C_ A /\ I = ( B |` dom I ) /\ (Edg ` S ) C_ ~P V ) ) )
24	23	3adant2 1043	. . . . 5 |- ( ( G e. W /\ Fun B /\ S e. UHGraph ) -> ( S SubGraph G <-> ( V C_ A /\ I = ( B |` dom I ) /\ (Edg ` S ) C_ ~P V ) ) )
25	24	adantr 276	. . . 4 |- ( ( ( G e. W /\ Fun B /\ S e. UHGraph ) /\ ( V C_ A /\ I C_ B ) ) -> ( S SubGraph G <-> ( V C_ A /\ I = ( B |` dom I ) /\ (Edg ` S ) C_ ~P V ) ) )
26	9, 14, 22, 25	mpbir3and 1207	. . 3 |- ( ( ( G e. W /\ Fun B /\ S e. UHGraph ) /\ ( V C_ A /\ I C_ B ) ) -> S SubGraph G )
27	26	ex 115	. 2 |- ( ( G e. W /\ Fun B /\ S e. UHGraph ) -> ( ( V C_ A /\ I C_ B ) -> S SubGraph G ) )
28	8, 27	impbid2 143	1 |- ( ( G e. W /\ Fun B /\ S e. UHGraph ) -> ( S SubGraph G <-> ( V C_ A /\ I C_ B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   C_ wss 3213   ~Pcpw 3671   class class class wbr 4111   dom cdm 4751   |` cres 4753   Fun wfun 5348   ` cfv 5354  Vtxcvtx 16024  iEdgciedg 16025  Edgcedg 16069  UHGraphcuhgr 16079   SubGraph csubgr 16265

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fo 5360  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-sub 8448  df-inn 9240  df-2 9298  df-3 9299  df-4 9300  df-5 9301  df-6 9302  df-7 9303  df-8 9304  df-9 9305  df-n0 9499  df-dec 9713  df-ndx 13232  df-slot 13233  df-base 13235  df-edgf 16017  df-vtx 16026  df-iedg 16027  df-edg 16070  df-uhgrm 16081  df-subgr 16266

This theorem is referenced by:  uhgrsubgrself  16278