Theorem subgruhgredgdm 16265

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 2296	. . 3 |- ( s = ( I ` X ) -> ( j e. s <-> j e. ( I ` X ) ) )
2	1	exbidv 1874	. 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 2232	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
6		subgruhgredgd.i	. . . . . 6 |- I = (iEdg ` S )
7		eqid 2232	. . . . . 6 |- (iEdg ` G ) = (iEdg ` G )
8		eqid 2232	. . . . . 6 |- (Edg ` S ) = (Edg ` S )
9	4, 5, 6, 7, 8	subgrprop2 16255	. . . . 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 1038	. . 3 |- ( ph -> (Edg ` S ) C_ ~P V )
12		subgruhgredgd.g	. . . . . 6 |- ( ph -> G e. UHGraph )
13		subgruhgrfun 16263	. . . . . 6 |- ( ( G e. UHGraph /\ S SubGraph G ) -> Fun (iEdg ` S ) )
14	12, 3, 13	syl2anc 411	. . . . 5 |- ( ph -> Fun (iEdg ` S ) )
15		subgruhgredgd.x	. . . . . 6 |- ( ph -> X e. dom I )
16	6	dmeqi 4957	. . . . . 6 |- dom I = dom (iEdg ` S )
17	15, 16	eleqtrdi 2325	. . . . 5 |- ( ph -> X e. dom (iEdg ` S ) )
18	6	fveq1i 5671	. . . . . 6 |- ( I ` X ) = ( (iEdg ` S ) ` X )
19		fvelrn 5808	. . . . . 6 |- ( ( Fun (iEdg ` S ) /\ X e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` X ) e. ran (iEdg ` S ) )
20	18, 19	eqeltrid 2319	. . . . 5 |- ( ( Fun (iEdg ` S ) /\ X e. dom (iEdg ` S ) ) -> ( I ` X ) e. ran (iEdg ` S ) )
21	14, 17, 20	syl2anc 411	. . . 4 |- ( ph -> ( I ` X ) e. ran (iEdg ` S ) )
22		edgval 16055	. . . 4 |- (Edg ` S ) = ran (iEdg ` S )
23	21, 22	eleqtrrdi 2326	. . 3 |- ( ph -> ( I ` X ) e. (Edg ` S ) )
24	11, 23	sseldd 3239	. 2 |- ( ph -> ( I ` X ) e. ~P V )
25	7	uhgrfun 16072	. . . . . 6 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
26	12, 25	syl 14	. . . . 5 |- ( ph -> Fun (iEdg ` G ) )
27	26	funfnd 5383	. . . 4 |- ( ph -> (iEdg ` G ) Fn dom (iEdg ` G ) )
28		subgreldmiedg 16264	. . . . 5 |- ( ( S SubGraph G /\ X e. dom (iEdg ` S ) ) -> X e. dom (iEdg ` G ) )
29	3, 17, 28	syl2anc 411	. . . 4 |- ( ph -> X e. dom (iEdg ` G ) )
30	7	uhgrm 16073	. . . 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 1274	. . 3 |- ( ph -> E. j j e. ( (iEdg ` G ) ` X ) )
32	10	simp2d 1037	. . . . . 6 |- ( ph -> I C_ (iEdg ` G ) )
33		funssfv 5696	. . . . . . 7 |- ( ( Fun (iEdg ` G ) /\ I C_ (iEdg ` G ) /\ X e. dom I ) -> ( (iEdg ` G ) ` X ) = ( I ` X ) )
34	33	eqcomd 2238	. . . . . 6 |- ( ( Fun (iEdg ` G ) /\ I C_ (iEdg ` G ) /\ X e. dom I ) -> ( I ` X ) = ( (iEdg ` G ) ` X ) )
35	26, 32, 15, 34	syl3anc 1274	. . . . 5 |- ( ph -> ( I ` X ) = ( (iEdg ` G ) ` X ) )
36	35	eleq2d 2302	. . . 4 |- ( ph -> ( j e. ( I ` X ) <-> j e. ( (iEdg ` G ) ` X ) ) )
37	36	exbidv 1874	. . 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 2975	1 |- ( ph -> ( I ` X ) e. { s e. ~P V | E. j j e. s } )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2203   {crab 2524   C_ wss 3211   ~Pcpw 3669   class class class wbr 4109   dom cdm 4749   ran crn 4750   Fun wfun 5346   Fn wfn 5347   ` cfv 5352  Vtxcvtx 16007  iEdgciedg 16008  Edgcedg 16052  UHGraphcuhgr 16062   SubGraph csubgr 16248

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

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 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-mpt 4173  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-fo 5358  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-sub 8446  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-dec 9710  df-ndx 13215  df-slot 13216  df-base 13218  df-edgf 16000  df-vtx 16009  df-iedg 16010  df-edg 16053  df-uhgrm 16064  df-subgr 16249

This theorem is referenced by:  subumgredg2en  16266  subuhgr  16267  subupgr  16268