Theorem uhgrissubgr 26167

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 2622	. . . 4 |- (Edg ` S ) = (Edg ` S )
6	1, 2, 3, 4, 5	subgrprop2 26166	. . 3 |- ( S SubGraph G -> ( V C_ A /\ I C_ B /\ (Edg ` S ) C_ ~P V ) )
7		3simpa 1058	. . 3 |- ( ( V C_ A /\ I C_ B /\ (Edg ` S ) C_ ~P V ) -> ( V C_ A /\ I C_ B ) )
8	6, 7	syl 17	. 2 |- ( S SubGraph G -> ( V C_ A /\ I C_ B ) )
9		simprl 794	. . . 4 |- ( ( ( G e. W /\ Fun B /\ S e. UHGraph ) /\ ( V C_ A /\ I C_ B ) ) -> V C_ A )
10		simp2 1062	. . . . . 6 |- ( ( G e. W /\ Fun B /\ S e. UHGraph ) -> Fun B )
11		simpr 477	. . . . . 6 |- ( ( V C_ A /\ I C_ B ) -> I C_ B )
12		funssres 5930	. . . . . 6 |- ( ( Fun B /\ I C_ B ) -> ( B |` dom I ) = I )
13	10, 11, 12	syl2an 494	. . . . 5 |- ( ( ( G e. W /\ Fun B /\ S e. UHGraph ) /\ ( V C_ A /\ I C_ B ) ) -> ( B |` dom I ) = I )
14	13	eqcomd 2628	. . . 4 |- ( ( ( G e. W /\ Fun B /\ S e. UHGraph ) /\ ( V C_ A /\ I C_ B ) ) -> I = ( B |` dom I ) )
15		edguhgr 26024	. . . . . . . . 9 |- ( ( S e. UHGraph /\ e e. (Edg ` S ) ) -> e e. ~P (Vtx ` S ) )
16	15	ex 450	. . . . . . . 8 |- ( S e. UHGraph -> ( e e. (Edg ` S ) -> e e. ~P (Vtx ` S ) ) )
17	1	pweqi 4162	. . . . . . . . 9 |- ~P V = ~P (Vtx ` S )
18	17	eleq2i 2693	. . . . . . . 8 |- ( e e. ~P V <-> e e. ~P (Vtx ` S ) )
19	16, 18	syl6ibr 242	. . . . . . 7 |- ( S e. UHGraph -> ( e e. (Edg ` S ) -> e e. ~P V ) )
20	19	ssrdv 3609	. . . . . 6 |- ( S e. UHGraph -> (Edg ` S ) C_ ~P V )
21	20	3ad2ant3 1084	. . . . 5 |- ( ( G e. W /\ Fun B /\ S e. UHGraph ) -> (Edg ` S ) C_ ~P V )
22	21	adantr 481	. . . 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 26163	. . . . . 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 1080	. . . . 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 481	. . . 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 1245	. . 3 |- ( ( ( G e. W /\ Fun B /\ S e. UHGraph ) /\ ( V C_ A /\ I C_ B ) ) -> S SubGraph G )
27	26	ex 450	. 2 |- ( ( G e. W /\ Fun B /\ S e. UHGraph ) -> ( ( V C_ A /\ I C_ B ) -> S SubGraph G ) )
28	8, 27	impbid2 216	1 |- ( ( G e. W /\ Fun B /\ S e. UHGraph ) -> ( S SubGraph G <-> ( V C_ A /\ I C_ B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   dom cdm 5114   |` cres 5116   Fun wfun 5882   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875  Edgcedg 25939   UHGraph cuhgr 25951   SubGraph csubgr 26159

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-edg 25940  df-uhgr 25953  df-subgr 26160

This theorem is referenced by:  uhgrsubgrself  26172