Theorem subgruhgredgdm 16194

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 2295	. . 3 |- ( s = ( I ` X ) -> ( j e. s <-> j e. ( I ` X ) ) )
2	1	exbidv 1873	. 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 2231	. . . . . 6 |- (Vtx ` G ) = (Vtx ` G )
6		subgruhgredgd.i	. . . . . 6 |- I = (iEdg ` S )
7		eqid 2231	. . . . . 6 |- (iEdg ` G ) = (iEdg ` G )
8		eqid 2231	. . . . . 6 |- (Edg ` S ) = (Edg ` S )
9	4, 5, 6, 7, 8	subgrprop2 16184	. . . . 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 16192	. . . . . 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 4938	. . . . . 6 |- dom I = dom (iEdg ` S )
17	15, 16	eleqtrdi 2324	. . . . 5 |- ( ph -> X e. dom (iEdg ` S ) )
18	6	fveq1i 5649	. . . . . 6 |- ( I ` X ) = ( (iEdg ` S ) ` X )
19		fvelrn 5786	. . . . . 6 |- ( ( Fun (iEdg ` S ) /\ X e. dom (iEdg ` S ) ) -> ( (iEdg ` S ) ` X ) e. ran (iEdg ` S ) )
20	18, 19	eqeltrid 2318	. . . . 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 15984	. . . 4 |- (Edg ` S ) = ran (iEdg ` S )
23	21, 22	eleqtrrdi 2325	. . 3 |- ( ph -> ( I ` X ) e. (Edg ` S ) )
24	11, 23	sseldd 3229	. 2 |- ( ph -> ( I ` X ) e. ~P V )
25	7	uhgrfun 16001	. . . . . 6 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
26	12, 25	syl 14	. . . . 5 |- ( ph -> Fun (iEdg ` G ) )
27	26	funfnd 5364	. . . 4 |- ( ph -> (iEdg ` G ) Fn dom (iEdg ` G ) )
28		subgreldmiedg 16193	. . . . 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 16002	. . . 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 5674	. . . . . . 7 |- ( ( Fun (iEdg ` G ) /\ I C_ (iEdg ` G ) /\ X e. dom I ) -> ( (iEdg ` G ) ` X ) = ( I ` X ) )
34	33	eqcomd 2237	. . . . . 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 2301	. . . 4 |- ( ph -> ( j e. ( I ` X ) <-> j e. ( (iEdg ` G ) ` X ) ) )
37	36	exbidv 1873	. . 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 2965	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 2202   {crab 2515   C_ wss 3201   ~Pcpw 3656   class class class wbr 4093   dom cdm 4731   ran crn 4732   Fun wfun 5327   Fn wfn 5328   ` cfv 5333  Vtxcvtx 15936  iEdgciedg 15937  Edgcedg 15981  UHGraphcuhgr 15991   SubGraph csubgr 16177

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-sub 8394  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-dec 9656  df-ndx 13148  df-slot 13149  df-base 13151  df-edgf 15929  df-vtx 15938  df-iedg 15939  df-edg 15982  df-uhgrm 15993  df-subgr 16178

This theorem is referenced by:  subumgredg2en  16195  subuhgr  16196  subupgr  16197