Theorem uhgrissubgr 16111

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 2231	. . . 4 |- (Edg ` S ) = (Edg ` S )
6	1, 2, 3, 4, 5	subgrprop2 16110	. . 3 |- ( S SubGraph G -> ( V C_ A /\ I C_ B /\ (Edg ` S ) C_ ~P V ) )
7		3simpa 1020	. . 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 1024	. . . . . 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 5369	. . . . . 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 2237	. . . 4 |- ( ( ( G e. W /\ Fun B /\ S e. UHGraph ) /\ ( V C_ A /\ I C_ B ) ) -> I = ( B |` dom I ) )
15		edguhgr 15987	. . . . . . . . 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 3656	. . . . . . . . 9 |- ~P V = ~P (Vtx ` S )
18	17	eleq2i 2298	. . . . . . . 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 3233	. . . . . 6 |- ( S e. UHGraph -> (Edg ` S ) C_ ~P V )
21	20	3ad2ant3 1046	. . . . 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 16107	. . . . . 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 1042	. . . . 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 1206	. . 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 1004   = wceq 1397   e. wcel 2202   C_ wss 3200   ~Pcpw 3652   class class class wbr 4088   dom cdm 4725   |` cres 4727   Fun wfun 5320   ` cfv 5326  Vtxcvtx 15862  iEdgciedg 15863  Edgcedg 15907  UHGraphcuhgr 15917   SubGraph csubgr 16103

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 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142

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-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fo 5332  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-sub 8351  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-5 9204  df-6 9205  df-7 9206  df-8 9207  df-9 9208  df-n0 9402  df-dec 9611  df-ndx 13084  df-slot 13085  df-base 13087  df-edgf 15855  df-vtx 15864  df-iedg 15865  df-edg 15908  df-uhgrm 15919  df-subgr 16104

This theorem is referenced by:  uhgrsubgrself  16116