Theorem subgruhgredgdm 16120

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 16110	. . . . 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 1037	. . 3 |- ( ph -> (Edg ` S ) C_ ~P V )
12		subgruhgredgd.g	. . . . . 6 |- ( ph -> G e. UHGraph )
13		subgruhgrfun 16118	. . . . . 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 4932	. . . . . 6 |- dom I = dom (iEdg ` S )
17	15, 16	eleqtrdi 2324	. . . . 5 |- ( ph -> X e. dom (iEdg ` S ) )
18	6	fveq1i 5640	. . . . . 6 |- ( I ` X ) = ( (iEdg ` S ) ` X )
19		fvelrn 5778	. . . . . 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 15910	. . . 4 |- (Edg ` S ) = ran (iEdg ` S )
23	21, 22	eleqtrrdi 2325	. . 3 |- ( ph -> ( I ` X ) e. (Edg ` S ) )
24	11, 23	sseldd 3228	. 2 |- ( ph -> ( I ` X ) e. ~P V )
25	7	uhgrfun 15927	. . . . . 6 |- ( G e. UHGraph -> Fun (iEdg ` G ) )
26	12, 25	syl 14	. . . . 5 |- ( ph -> Fun (iEdg ` G ) )
27	26	funfnd 5357	. . . 4 |- ( ph -> (iEdg ` G ) Fn dom (iEdg ` G ) )
28		subgreldmiedg 16119	. . . . 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 15928	. . . 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 1273	. . 3 |- ( ph -> E. j j e. ( (iEdg ` G ) ` X ) )
32	10	simp2d 1036	. . . . . 6 |- ( ph -> I C_ (iEdg ` G ) )
33		funssfv 5665	. . . . . . 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 1273	. . . . 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 2964	1 |- ( ph -> ( I ` X ) e. { s e. ~P V | E. j j e. s } )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1004   = wceq 1397   E.wex 1540   e. wcel 2202   {crab 2514   C_ wss 3200   ~Pcpw 3652   class class class wbr 4088   dom cdm 4725   ran crn 4726   Fun wfun 5320   Fn wfn 5321   ` 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-ima 4738  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:  subumgredg2en  16121  subuhgr  16122  subupgr  16123