Theorem usgrstrrepeen 16213

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 16033	. . . . . . . 8 |- ( ph -> (Vtx ` ( G sSet <. I , E >. ) ) = ( Base ` G ) )
7		usgrstrrepe.v	. . . . . . . 8 |- V = ( Base ` G )
8	6, 7	eqtr4di 2283	. . . . . . 7 |- ( ph -> (Vtx ` ( G sSet <. I , E >. ) ) = V )
9	8	pweqd 3673	. . . . . 6 |- ( ph -> ~P (Vtx ` ( G sSet <. I , E >. ) ) = ~P V )
10	9	rabeqdv 2806	. . . . 5 |- ( ph -> { x e. ~P (Vtx ` ( G sSet <. I , E >. ) ) | x ~~ 2o } = { x e. ~P V | x ~~ 2o } )
11		f1eq3 5569	. . . . 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 16034	. . . 4 |- ( ph -> (iEdg ` ( G sSet <. I , E >. ) ) = E )
15	14	dmeqd 4957	. . . 4 |- ( ph -> dom (iEdg ` ( G sSet <. I , E >. ) ) = dom E )
16		eqidd 2233	. . . 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 5605	. . 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 13213	. . . . 5 |- ( G Struct X -> G e. _V )
20	3, 19	syl 14	. . . 4 |- ( ph -> G e. _V )
21		edgfndxnn 15990	. . . . . 6 |- (.ef ` ndx ) e. NN
22	2, 21	eqeltri 2305	. . . . 5 |- I e. NN
23	22	a1i 9	. . . 4 |- ( ph -> I e. NN )
24		setsex 13233	. . . 4 |- ( ( G e. _V /\ I e. NN /\ E e. W ) -> ( G sSet <. I , E >. ) e. _V )
25	20, 23, 5, 24	syl3anc 1274	. . 3 |- ( ph -> ( G sSet <. I , E >. ) e. _V )
26		eqid 2232	. . . 4 |- (Vtx ` ( G sSet <. I , E >. ) ) = (Vtx ` ( G sSet <. I , E >. ) )
27		eqid 2232	. . . 4 |- (iEdg ` ( G sSet <. I , E >. ) ) = (iEdg ` ( G sSet <. I , E >. ) )
28	26, 27	isusgren 16140	. . 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 1398   e. wcel 2203   {crab 2524   _Vcvv 2812   ~Pcpw 3668   <.cop 3691   class class class wbr 4108   dom cdm 4748   -1-1->wf1 5348   ` cfv 5351  (class class class)co 6049   2oc2o 6640   ~~ cen 6972   NNcn 9233   Struct cstr 13197   ndxcnx 13198   sSet csts 13199   Basecbs 13201  .efcedgf 15986  Vtxcvtx 15994  iEdgciedg 15995  USGraphcusgr 16136

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 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-ltadd 8239  ax-pre-mulgt0 8240

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  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-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-iord 4486  df-on 4488  df-suc 4491  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-1o 6646  df-2o 6647  df-en 6975  df-dom 6976  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-inn 9234  df-2 9292  df-3 9293  df-4 9294  df-5 9295  df-6 9296  df-7 9297  df-8 9298  df-9 9299  df-n0 9493  df-z 9574  df-dec 9706  df-struct 13203  df-ndx 13204  df-slot 13205  df-base 13207  df-sets 13208  df-edgf 15987  df-vtx 15996  df-iedg 15997  df-usgren 16138

This theorem is referenced by: (None)