Theorem wrdupgren 16203

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 16202	. . 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 11255	. . . . 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 5520	. . 3 |- ( ( G e. U /\ E e. Word X ) -> dom E = ( 0..^ (♯ ` E ) ) )
8	7	feq2d 5501	. 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 11253	. . . . . 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 11254	. . . . 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 11255	. . 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 716   = wceq 1398   e. wcel 2205   {crab 2526   ~Pcpw 3674   class class class wbr 4114   dom cdm 4754   -->wf 5353   ` cfv 5357  (class class class)co 6058   1oc1o 6653   2oc2o 6654   ~~ cen 6986   0cc0 8143   NN0cn0 9513  ..^cfzo 10498  ♯chash 11163  Word cword 11249  Vtxcvtx 16119  iEdgciedg 16120  UPGraphcupgr 16198

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 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-1o 6660  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-z 9595  df-dec 9728  df-uz 9872  df-fz 10362  df-fzo 10499  df-ihash 11164  df-word 11250  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16112  df-vtx 16121  df-iedg 16122  df-upgren 16200

This theorem is referenced by:  vdegp1aid  16421  vdegp1bid  16422