Theorem subgruhgredgdm 16425

Description: An edge of a subgraph of a hypergraph is an inhabited subset of its vertices. (Contributed by AV, 17-Nov-2020.) (Revised by AV, 21-Nov-2020.)

Hypotheses

Ref	Expression
subgruhgredgd.v	|- V = (Vtx ` S )
subgruhgredgd.i	|- I = (iEdg ` S )
subgruhgredgd.g	|- ( ph -> G e. UHGraph )
subgruhgredgd.s	|- ( ph -> S SubGraph G )
subgruhgredgd.x	|- ( ph -> X e. dom I )

Assertion

Ref	Expression
subgruhgredgdm	|- ( ph -> ( I ` X ) e. { s e. ~P V | E. j j e. s } )

Distinct variable groups:   j, G   j, I, s   V, s   j, X, s   ph, j

Allowed substitution hints:   ph( s)   S( j, s)   G( s)   V( j)

Proof of Theorem subgruhgredgdm

Step	Hyp	Ref	Expression
1		eleq2 2302	. . 3 |- ( s = ( I ` X ) -> ( j e. s <-> j e. ( I ` X ) ) )
2	1	exbidv 1878	. 2 |- ( s = ( I ` X ) -> ( E. j j e. s <-> E. j j e. ( I ` X ) ) )
3		subgruhgredgd.s	. . . . 5 |- ( ph -> S SubGraph G )
4		subgruhgredgd.v	. . . . . 6 |- V = (Vtx ` S )
5		eqid 2238	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
6		subgruhgredgd.i	. . . . . 6 |- I = (iEdg ` S )
7		eqid 2238	. . . . . 6 |- (iEdg ` G ) = (iEdg ` G )
8		eqid 2238	. . . . . 6 |- (Edg ` S ) = (Edg ` S )
9	4, 5, 6, 7, 8	subgrprop2 16415	. . . . 5 |- ( S SubGraph G -> ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) )
10	3, 9	syl 14	. . . 4 |- ( ph -> ( V C_ (Vtx ` G ) /\ I C_ (iEdg ` G ) /\ (Edg ` S ) C_ ~P V ) )
11	10	simp3d 1042	. . 3 |- ( ph -> (Edg ` S ) C_ ~P V )
12		subgruhgredgd.g	. . . . . 6 |- ( ph -> G e. UHGraph )
13		subgruhgrfun 16423	. . . . . 6 |- ( ( G e. UHGraph /\ S SubGraph G ) -> Fun (iEdg ` S ) )
14	12, 3, 13	syl2anc 415	. . . . 5 |- ( ph -> Fun (iEdg ` S ) )
15		subgruhgredgd.x	. . . . . 6 |- ( ph -> X e. dom I )
16	6	dmeqi 4977	. . . . . 6 |- dom I = dom (iEdg ` S )
17	15, 16	eleqtrdi 2331	. . . . 5 |- ( ph -> X e. dom (iEdg ` S ) )
18	6	fveq1i 5691	. . . . . 6 |- ( I ` X ) = ( (iEdg ` S ) ` X )
19		fvelrn 5830	. . . . . 6 |- ( ( Fun (iEdg ` S ) /\ X e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` X ) e. ran (iEdg ` S ) )
20	18, 19	eqeltrid 2325	. . . . 5 |- ( ( Fun (iEdg ` S ) /\ X e. dom (iEdg ` S ) ) -> ( I ` X ) e. ran (iEdg ` S ) )
21	14, 17, 20	syl2anc 415	. . . 4 |- ( ph -> ( I ` X ) e. ran (iEdg ` S ) )
22		edgval 16215	. . . 4 |- (Edg ` S ) = ran (iEdg ` S )
23	21, 22	eleqtrrdi 2332	. . 3 |- ( ph -> ( I ` X ) e. (Edg ` S ) )
24	11, 23	sseldd 3249	. 2 |- ( ph -> ( I ` X ) e. ~P V )
25	7	uhgrfun 16232	. . . . . 6 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
26	12, 25	syl 14	. . . . 5 |- ( ph -> Fun (iEdg ` G ) )
27	26	funfnd 5403	. . . 4 |- ( ph -> (iEdg ` G ) Fn dom (iEdg ` G ) )
28		subgreldmiedg 16424	. . . . 5 |- ( ( S SubGraph G /\ X e. dom (iEdg ` S ) ) -> X e. dom (iEdg ` G ) )
29	3, 17, 28	syl2anc 415	. . . 4 |- ( ph -> X e. dom (iEdg ` G ) )
30	7	uhgrm 16233	. . . 4 |- ( ( G e. UHGraph /\ (iEdg ` G ) Fn dom (iEdg ` G ) /\ X e. dom (iEdg ` G ) ) -> E. j j e. ( (iEdg ` G ) ` X ) )
31	12, 27, 29, 30	syl3anc 1278	. . 3 |- ( ph -> E. j j e. ( (iEdg ` G ) ` X ) )
32	10	simp2d 1041	. . . . . 6 |- ( ph -> I C_ (iEdg ` G ) )
33		funssfv 5716	. . . . . . 7 |- ( ( Fun (iEdg ` G ) /\ I C_ (iEdg ` G ) /\ X e. dom I ) -> ( (iEdg ` G ) ` X ) = ( I ` X ) )
34	33	eqcomd 2244	. . . . . 6 |- ( ( Fun (iEdg ` G ) /\ I C_ (iEdg ` G ) /\ X e. dom I ) -> ( I ` X ) = ( (iEdg ` G ) ` X ) )
35	26, 32, 15, 34	syl3anc 1278	. . . . 5 |- ( ph -> ( I ` X ) = ( (iEdg ` G ) ` X ) )
36	35	eleq2d 2308	. . . 4 |- ( ph -> ( j e. ( I ` X ) <-> j e. ( (iEdg ` G ) ` X ) ) )
37	36	exbidv 1878	. . 3 |- ( ph -> ( E. j j e. ( I ` X ) <-> E. j j e. ( (iEdg ` G ) ` X ) ) )
38	31, 37	mpbird 167	. 2 |- ( ph -> E. j j e. ( I ` X ) )
39	2, 24, 38	elrabd 2984	1 |- ( ph -> ( I ` X ) e. { s e. ~P V | E. j j e. s } )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   E.wex 1545   e. wcel 2209   {crab 2532   C_ wss 3220   ~Pcpw 3685   class class class wbr 4125   dom cdm 4769   ran crn 4770   Fun wfun 5366   Fn wfn 5367   ` 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-ima 4782  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:  subumgredg2en  16426  subuhgr  16427  subupgr  16428