Theorem uhgrissubgr 16416

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 2238	. . . 4 |- (Edg ` S ) = (Edg ` S )
6	1, 2, 3, 4, 5	subgrprop2 16415	. . 3 |- ( S SubGraph G -> ( V C_ A /\ I C_ B /\ (Edg ` S ) C_ ~P V ) )
7		3simpa 1025	. . 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 535	. . . 4 |- ( ( ( G e. W /\ Fun B /\ S e. UHGraph ) /\ ( V C_ A /\ I C_ B ) ) -> V C_ A )
10		simp2 1029	. . . . . 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 5415	. . . . . 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 2244	. . . 4 |- ( ( ( G e. W /\ Fun B /\ S e. UHGraph ) /\ ( V C_ A /\ I C_ B ) ) -> I = ( B |` dom I ) )
15		edguhgr 16292	. . . . . . . . 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 3689	. . . . . . . . 9 |- ~P V = ~P (Vtx ` S )
18	17	eleq2i 2305	. . . . . . . 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 3254	. . . . . 6 |- ( S e. UHGraph -> (Edg ` S ) C_ ~P V )
21	20	3ad2ant3 1051	. . . . 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 16412	. . . . . 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 1047	. . . . 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 1211	. . 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 1009   = wceq 1402   e. wcel 2209   C_ wss 3220   ~Pcpw 3685   class class class wbr 4125   dom cdm 4769   |` cres 4771   Fun wfun 5366   ` cfv 5372  Vtxcvtx 16167  iEdgciedg 16168  Edgcedg 16212  UHGraphcuhgr 16222   SubGraph csubgr 16408

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 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 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fo 5378  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-sub 8489  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-5 9345  df-6 9346  df-7 9347  df-8 9348  df-9 9349  df-n0 9543  df-dec 9757  df-ndx 13333  df-slot 13334  df-base 13336  df-edgf 16160  df-vtx 16169  df-iedg 16170  df-edg 16213  df-uhgrm 16224  df-subgr 16409

This theorem is referenced by:  uhgrsubgrself  16421