Theorem usgrstrrepeen 16111

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 15931	. . . . . . . 8 |- ( ph -> (Vtx ` ( G sSet <. I , E >. ) ) = ( Base ` G ) )
7		usgrstrrepe.v	. . . . . . . 8 |- V = ( Base ` G )
8	6, 7	eqtr4di 2281	. . . . . . 7 |- ( ph -> (Vtx ` ( G sSet <. I , E >. ) ) = V )
9	8	pweqd 3658	. . . . . 6 |- ( ph -> ~P (Vtx ` ( G sSet <. I , E >. ) ) = ~P V )
10	9	rabeqdv 2795	. . . . 5 |- ( ph -> { x e. ~P (Vtx ` ( G sSet <. I , E >. ) ) | x ~~ 2o } = { x e. ~P V | x ~~ 2o } )
11		f1eq3 5542	. . . . 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 15932	. . . 4 |- ( ph -> (iEdg ` ( G sSet <. I , E >. ) ) = E )
15	14	dmeqd 4935	. . . 4 |- ( ph -> dom (iEdg ` ( G sSet <. I , E >. ) ) = dom E )
16		eqidd 2231	. . . 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 5578	. . 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 13117	. . . . 5 |- ( G Struct X -> G e. _V )
20	3, 19	syl 14	. . . 4 |- ( ph -> G e. _V )
21		edgfndxnn 15888	. . . . . 6 |- (.ef ` ndx ) e. NN
22	2, 21	eqeltri 2303	. . . . 5 |- I e. NN
23	22	a1i 9	. . . 4 |- ( ph -> I e. NN )
24		setsex 13137	. . . 4 |- ( ( G e. _V /\ I e. NN /\ E e. W ) -> ( G sSet <. I , E >. ) e. _V )
25	20, 23, 5, 24	syl3anc 1273	. . 3 |- ( ph -> ( G sSet <. I , E >. ) e. _V )
26		eqid 2230	. . . 4 |- (Vtx ` ( G sSet <. I , E >. ) ) = (Vtx ` ( G sSet <. I , E >. ) )
27		eqid 2230	. . . 4 |- (iEdg ` ( G sSet <. I , E >. ) ) = (iEdg ` ( G sSet <. I , E >. ) )
28	26, 27	isusgren 16038	. . 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 1397   e. wcel 2201   {crab 2513   _Vcvv 2801   ~Pcpw 3653   <.cop 3673   class class class wbr 4089   dom cdm 4727   -1-1->wf1 5325   ` cfv 5328  (class class class)co 6023   2oc2o 6581   ~~ cen 6912   NNcn 9148   Struct cstr 13101   ndxcnx 13102   sSet csts 13103   Basecbs 13105  .efcedgf 15884  Vtxcvtx 15892  iEdgciedg 15893  USGraphcusgr 16034

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 2203  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-mulrcl 8136  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-precex 8147  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-ltadd 8153  ax-pre-mulgt0 8154

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-iord 4465  df-on 4467  df-suc 4470  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-1o 6587  df-2o 6588  df-en 6915  df-dom 6916  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-inn 9149  df-2 9207  df-3 9208  df-4 9209  df-5 9210  df-6 9211  df-7 9212  df-8 9213  df-9 9214  df-n0 9408  df-z 9485  df-dec 9617  df-struct 13107  df-ndx 13108  df-slot 13109  df-base 13111  df-sets 13112  df-edgf 15885  df-vtx 15894  df-iedg 15895  df-usgren 16036

This theorem is referenced by: (None)