Theorem wrdupgren 15881

Description: The property of being an undirected pseudograph, expressing the edges as "words". (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)

Hypotheses

Ref	Expression
isupgr.v	|- V = (Vtx ` G )
isupgr.e	|- E = (iEdg ` G )

Assertion

Ref	Expression
wrdupgren	|- ( ( G e. U /\ E e. Word X ) -> ( G e. UPGraph <-> E e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) )

Distinct variable groups:   x, G   x, V

Allowed substitution hints:   U( x)   E( x)   X( x)

Proof of Theorem wrdupgren

Step	Hyp	Ref	Expression
1		isupgr.v	. . . 4 |- V = (Vtx ` G )
2		isupgr.e	. . . 4 |- E = (iEdg ` G )
3	1, 2	isupgren 15880	. . 3 |- ( G e. U -> ( G e. UPGraph <-> E : dom E --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) )
4	3	adantr 276	. 2 |- ( ( G e. U /\ E e. Word X ) -> ( G e. UPGraph <-> E : dom E --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) )
5		wrdf 11064	. . . . 5 |- ( E e. Word X -> E : ( 0..^ (♯ ` E ) ) --> X )
6	5	adantl 277	. . . 4 |- ( ( G e. U /\ E e. Word X ) -> E : ( 0..^ (♯ ` E ) ) --> X )
7	6	fdmd 5476	. . 3 |- ( ( G e. U /\ E e. Word X ) -> dom E = ( 0..^ (♯ ` E ) ) )
8	7	feq2d 5457	. 2 |- ( ( G e. U /\ E e. Word X ) -> ( E : dom E --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } <-> E : ( 0..^ (♯ ` E ) ) --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) )
9		simpr 110	. . . . 5 |- ( ( ( G e. U /\ E e. Word X ) /\ E : ( 0..^ (♯ ` E ) ) --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) -> E : ( 0..^ (♯ ` E ) ) --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
10		lencl 11062	. . . . . 6 |- ( E e. Word X -> (♯ ` E ) e. NN0 )
11	10	ad2antlr 489	. . . . 5 |- ( ( ( G e. U /\ E e. Word X ) /\ E : ( 0..^ (♯ ` E ) ) --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) -> (♯ ` E ) e. NN0 )
12		iswrdinn0 11063	. . . . 5 |- ( ( E : ( 0..^ (♯ ` E ) ) --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } /\ (♯ ` E ) e. NN0 ) -> E e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
13	9, 11, 12	syl2anc 411	. . . 4 |- ( ( ( G e. U /\ E e. Word X ) /\ E : ( 0..^ (♯ ` E ) ) --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) -> E e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
14	13	ex 115	. . 3 |- ( ( G e. U /\ E e. Word X ) -> ( E : ( 0..^ (♯ ` E ) ) --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } -> E e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) )
15		wrdf 11064	. . 3 |- ( E e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } -> E : ( 0..^ (♯ ` E ) ) --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
16	14, 15	impbid1 142	. 2 |- ( ( G e. U /\ E e. Word X ) -> ( E : ( 0..^ (♯ ` E ) ) --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } <-> E e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) )
17	4, 8, 16	3bitrd 214	1 |- ( ( G e. U /\ E e. Word X ) -> ( G e. UPGraph <-> E e. Word { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713   = wceq 1395   e. wcel 2200   {crab 2512   ~Pcpw 3649   class class class wbr 4082   dom cdm 4716   -->wf 5310   ` cfv 5314  (class class class)co 5994   1oc1o 6545   2oc2o 6546   ~~ cen 6875   0cc0 7987   NN0cn0 9357  ..^cfzo 10326  ♯chash 10984  Word cword 11058  Vtxcvtx 15798  iEdgciedg 15799  UPGraphcupgr 15876

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-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4521  ax-setind 4626  ax-iinf 4677  ax-cnex 8078  ax-resscn 8079  ax-1cn 8080  ax-1re 8081  ax-icn 8082  ax-addcl 8083  ax-addrcl 8084  ax-mulcl 8085  ax-addcom 8087  ax-mulcom 8088  ax-addass 8089  ax-mulass 8090  ax-distr 8091  ax-i2m1 8092  ax-0lt1 8093  ax-1rid 8094  ax-0id 8095  ax-rnegex 8096  ax-cnre 8098  ax-pre-ltirr 8099  ax-pre-ltwlin 8100  ax-pre-lttrn 8101  ax-pre-apti 8102  ax-pre-ltadd 8103

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 3888  df-int 3923  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4381  df-iord 4454  df-on 4456  df-ilim 4457  df-suc 4459  df-iom 4680  df-xp 4722  df-rel 4723  df-cnv 4724  df-co 4725  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729  df-iota 5274  df-fun 5316  df-fn 5317  df-f 5318  df-f1 5319  df-fo 5320  df-f1o 5321  df-fv 5322  df-riota 5947  df-ov 5997  df-oprab 5998  df-mpo 5999  df-1st 6276  df-2nd 6277  df-recs 6441  df-frec 6527  df-1o 6552  df-er 6670  df-en 6878  df-dom 6879  df-fin 6880  df-pnf 8171  df-mnf 8172  df-xr 8173  df-ltxr 8174  df-le 8175  df-sub 8307  df-neg 8308  df-inn 9099  df-2 9157  df-3 9158  df-4 9159  df-5 9160  df-6 9161  df-7 9162  df-8 9163  df-9 9164  df-n0 9358  df-z 9435  df-dec 9567  df-uz 9711  df-fz 10193  df-fzo 10327  df-ihash 10985  df-word 11059  df-ndx 13021  df-slot 13022  df-base 13024  df-edgf 15791  df-vtx 15800  df-iedg 15801  df-upgren 15878

This theorem is referenced by: (None)