Theorem usgrstrrepeen 16050

Description: Replacing (or adding) the edges (between elements of the base set) of an extensible structure results in a simple graph. Instead of requiring ( ph -> G Struct X ), it would be sufficient to require ( ph -> Fun ( G \ { (/) } ) ) and ( ph -> G e. _V ). (Contributed by AV, 13-Nov-2021.) (Proof shortened by AV, 16-Nov-2021.)

Hypotheses

Ref	Expression
usgrstrrepe.v	|- V = ( Base ` G )
usgrstrrepe.i	|- I = (.ef ` ndx )
usgrstrrepe.s	|- ( ph -> G Struct X )
usgrstrrepe.b	|- ( ph -> ( Base ` ndx ) e. dom G )
usgrstrrepe.w	|- ( ph -> E e. W )
usgrstrrepeen.e	|- ( ph -> E : dom E -1-1-> { x e. ~P V | x ~~ 2o } )

Assertion

Ref	Expression
usgrstrrepeen	|- ( ph -> ( G sSet <. I , E >. ) e. USGraph )

Distinct variable groups:   x, G   x, E   x, I   x, V   ph, x

Allowed substitution hints:   W( x)   X( x)

Proof of Theorem usgrstrrepeen

Step	Hyp	Ref	Expression
1		usgrstrrepeen.e	. . . 4 |- ( ph -> E : dom E -1-1-> { x e. ~P V | x ~~ 2o } )
2		usgrstrrepe.i	. . . . . . . . 9 |- I = (.ef ` ndx )
3		usgrstrrepe.s	. . . . . . . . 9 |- ( ph -> G Struct X )
4		usgrstrrepe.b	. . . . . . . . 9 |- ( ph -> ( Base ` ndx ) e. dom G )
5		usgrstrrepe.w	. . . . . . . . 9 |- ( ph -> E e. W )
6	2, 3, 4, 5	setsvtx 15873	. . . . . . . 8 |- ( ph -> (Vtx ` ( G sSet <. I , E >. ) ) = ( Base ` G ) )
7		usgrstrrepe.v	. . . . . . . 8 |- V = ( Base ` G )
8	6, 7	eqtr4di 2280	. . . . . . 7 |- ( ph -> (Vtx ` ( G sSet <. I , E >. ) ) = V )
9	8	pweqd 3654	. . . . . 6 |- ( ph -> ~P (Vtx ` ( G sSet <. I , E >. ) ) = ~P V )
10	9	rabeqdv 2793	. . . . 5 |- ( ph -> { x e. ~P (Vtx ` ( G sSet <. I , E >. ) ) | x ~~ 2o } = { x e. ~P V | x ~~ 2o } )
11		f1eq3 5533	. . . . 5 |- ( { x e. ~P (Vtx ` ( G sSet <. I , E >. ) ) | x ~~ 2o } = { x e. ~P V | x ~~ 2o } -> ( E : dom E -1-1-> { x e. ~P (Vtx ` ( G sSet <. I , E >. ) ) | x ~~ 2o } <-> E : dom E -1-1-> { x e. ~P V | x ~~ 2o } ) )
12	10, 11	syl 14	. . . 4 |- ( ph -> ( E : dom E -1-1-> { x e. ~P (Vtx ` ( G sSet <. I , E >. ) ) | x ~~ 2o } <-> E : dom E -1-1-> { x e. ~P V | x ~~ 2o } ) )
13	1, 12	mpbird 167	. . 3 |- ( ph -> E : dom E -1-1-> { x e. ~P (Vtx ` ( G sSet <. I , E >. ) ) | x ~~ 2o } )
14	2, 3, 4, 5	setsiedg 15874	. . . 4 |- ( ph -> (iEdg ` ( G sSet <. I , E >. ) ) = E )
15	14	dmeqd 4928	. . . 4 |- ( ph -> dom (iEdg ` ( G sSet <. I , E >. ) ) = dom E )
16		eqidd 2230	. . . 4 |- ( ph -> { x e. ~P (Vtx ` ( G sSet <. I , E >. ) ) | x ~~ 2o } = { x e. ~P (Vtx ` ( G sSet <. I , E >. ) ) | x ~~ 2o } )
17	14, 15, 16	f1eq123d 5569	. . 3 |- ( ph -> ( (iEdg ` ( G sSet <. I , E >. ) ) : dom (iEdg ` ( G sSet <. I , E >. ) ) -1-1-> { x e. ~P (Vtx ` ( G sSet <. I , E >. ) ) | x ~~ 2o } <-> E : dom E -1-1-> { x e. ~P (Vtx ` ( G sSet <. I , E >. ) ) | x ~~ 2o } ) )
18	13, 17	mpbird 167	. 2 |- ( ph -> (iEdg ` ( G sSet <. I , E >. ) ) : dom (iEdg ` ( G sSet <. I , E >. ) ) -1-1-> { x e. ~P (Vtx ` ( G sSet <. I , E >. ) ) | x ~~ 2o } )
19		structex 13065	. . . . 5 |- ( G Struct X -> G e. _V )
20	3, 19	syl 14	. . . 4 |- ( ph -> G e. _V )
21		edgfndxnn 15830	. . . . . 6 |- (.ef ` ndx ) e. NN
22	2, 21	eqeltri 2302	. . . . 5 |- I e. NN
23	22	a1i 9	. . . 4 |- ( ph -> I e. NN )
24		setsex 13085	. . . 4 |- ( ( G e. _V /\ I e. NN /\ E e. W ) -> ( G sSet <. I , E >. ) e. _V )
25	20, 23, 5, 24	syl3anc 1271	. . 3 |- ( ph -> ( G sSet <. I , E >. ) e. _V )
26		eqid 2229	. . . 4 |- (Vtx ` ( G sSet <. I , E >. ) ) = (Vtx ` ( G sSet <. I , E >. ) )
27		eqid 2229	. . . 4 |- (iEdg ` ( G sSet <. I , E >. ) ) = (iEdg ` ( G sSet <. I , E >. ) )
28	26, 27	isusgren 15977	. . 3 |- ( ( G sSet <. I , E >. ) e. _V -> ( ( G sSet <. I , E >. ) e. USGraph <-> (iEdg ` ( G sSet <. I , E >. ) ) : dom (iEdg ` ( G sSet <. I , E >. ) ) -1-1-> { x e. ~P (Vtx ` ( G sSet <. I , E >. ) ) | x ~~ 2o } ) )
29	25, 28	syl 14	. 2 |- ( ph -> ( ( G sSet <. I , E >. ) e. USGraph <-> (iEdg ` ( G sSet <. I , E >. ) ) : dom (iEdg ` ( G sSet <. I , E >. ) ) -1-1-> { x e. ~P (Vtx ` ( G sSet <. I , E >. ) ) | x ~~ 2o } ) )
30	18, 29	mpbird 167	1 |- ( ph -> ( G sSet <. I , E >. ) e. USGraph )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1395   e. wcel 2200   {crab 2512   _Vcvv 2799   ~Pcpw 3649   <.cop 3669   class class class wbr 4083   dom cdm 4720   -1-1->wf1 5318   ` cfv 5321  (class class class)co 6010   2oc2o 6567   ~~ cen 6898   NNcn 9126   Struct cstr 13049   ndxcnx 13050   sSet csts 13051   Basecbs 13053  .efcedgf 15826  Vtxcvtx 15834  iEdgciedg 15835  USGraphcusgr 15973

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-ltadd 8131  ax-pre-mulgt0 8132

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4385  df-iord 4458  df-on 4460  df-suc 4463  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-f1 5326  df-fo 5327  df-f1o 5328  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-1o 6573  df-2o 6574  df-en 6901  df-dom 6902  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-inn 9127  df-2 9185  df-3 9186  df-4 9187  df-5 9188  df-6 9189  df-7 9190  df-8 9191  df-9 9192  df-n0 9386  df-z 9463  df-dec 9595  df-struct 13055  df-ndx 13056  df-slot 13057  df-base 13059  df-sets 13060  df-edgf 15827  df-vtx 15836  df-iedg 15837  df-usgren 15975

This theorem is referenced by: (None)