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Theorem wrdl1s1 12674
Description: A word of length 1 is a singleton word consisting of the first symbol of the word. (Contributed by AV, 22-Jul-2018.) (Proof shortened by AV, 14-Oct-2018.)
Assertion
Ref Expression
wrdl1s1  |-  ( S  e.  V  ->  ( W  =  <" S ">  <->  ( W  e. Word  V  /\  ( # `  W
)  =  1  /\  ( W `  0
)  =  S ) ) )

Proof of Theorem wrdl1s1
StepHypRef Expression
1 s1cl 12666 . . . 4  |-  ( S  e.  V  ->  <" S ">  e. Word  V )
2 s1len 12669 . . . . 5  |-  ( # `  <" S "> )  =  1
32a1i 11 . . . 4  |-  ( S  e.  V  ->  ( # `
 <" S "> )  =  1
)
4 s1fv 12671 . . . 4  |-  ( S  e.  V  ->  ( <" S "> `  0 )  =  S )
51, 3, 43jca 1177 . . 3  |-  ( S  e.  V  ->  ( <" S ">  e. Word  V  /\  ( # `  <" S "> )  =  1  /\  ( <" S "> `  0 )  =  S ) )
6 eleq1 2474 . . . 4  |-  ( W  =  <" S ">  ->  ( W  e. Word  V 
<-> 
<" S ">  e. Word  V ) )
7 fveq2 5848 . . . . 5  |-  ( W  =  <" S ">  ->  ( # `  W
)  =  ( # `  <" S "> ) )
87eqeq1d 2404 . . . 4  |-  ( W  =  <" S ">  ->  ( ( # `  W )  =  1  <-> 
( # `  <" S "> )  =  1 ) )
9 fveq1 5847 . . . . 5  |-  ( W  =  <" S ">  ->  ( W ` 
0 )  =  (
<" S "> `  0 ) )
109eqeq1d 2404 . . . 4  |-  ( W  =  <" S ">  ->  ( ( W `
 0 )  =  S  <->  ( <" S "> `  0 )  =  S ) )
116, 8, 103anbi123d 1301 . . 3  |-  ( W  =  <" S ">  ->  ( ( W  e. Word  V  /\  ( # `
 W )  =  1  /\  ( W `
 0 )  =  S )  <->  ( <" S ">  e. Word  V  /\  ( # `  <" S "> )  =  1  /\  ( <" S "> `  0 )  =  S ) ) )
125, 11syl5ibrcom 222 . 2  |-  ( S  e.  V  ->  ( W  =  <" S ">  ->  ( W  e. Word  V  /\  ( # `  W )  =  1  /\  ( W ` 
0 )  =  S ) ) )
13 eqs1 12673 . . . 4  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  1 )  ->  W  =  <" ( W `  0 ) "> )
14 s1eq 12664 . . . . 5  |-  ( ( W `  0 )  =  S  ->  <" ( W `  0 ) ">  =  <" S "> )
1514eqeq2d 2416 . . . 4  |-  ( ( W `  0 )  =  S  ->  ( W  =  <" ( W `  0 ) ">  <->  W  =  <" S "> )
)
1613, 15syl5ibcom 220 . . 3  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  1 )  -> 
( ( W ` 
0 )  =  S  ->  W  =  <" S "> )
)
17163impia 1194 . 2  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  1  /\  ( W `  0 )  =  S )  ->  W  =  <" S "> )
1812, 17impbid1 203 1  |-  ( S  e.  V  ->  ( W  =  <" S ">  <->  ( W  e. Word  V  /\  ( # `  W
)  =  1  /\  ( W `  0
)  =  S ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ` cfv 5568   0cc0 9521   1c1 9522   #chash 12450  Word cword 12581   <"cs1 12584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573  ax-cnex 9577  ax-resscn 9578  ax-1cn 9579  ax-icn 9580  ax-addcl 9581  ax-addrcl 9582  ax-mulcl 9583  ax-mulrcl 9584  ax-mulcom 9585  ax-addass 9586  ax-mulass 9587  ax-distr 9588  ax-i2m1 9589  ax-1ne0 9590  ax-1rid 9591  ax-rnegex 9592  ax-rrecex 9593  ax-cnre 9594  ax-pre-lttri 9595  ax-pre-lttrn 9596  ax-pre-ltadd 9597  ax-pre-mulgt0 9598
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rmo 2761  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-pss 3429  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-tp 3976  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-br 4395  df-opab 4453  df-mpt 4454  df-tr 4489  df-eprel 4733  df-id 4737  df-po 4743  df-so 4744  df-fr 4781  df-we 4783  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-pred 5366  df-ord 5412  df-on 5413  df-lim 5414  df-suc 5415  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-riota 6239  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6683  df-1st 6783  df-2nd 6784  df-wrecs 7012  df-recs 7074  df-rdg 7112  df-1o 7166  df-oadd 7170  df-er 7347  df-en 7554  df-dom 7555  df-sdom 7556  df-fin 7557  df-card 8351  df-cda 8579  df-pnf 9659  df-mnf 9660  df-xr 9661  df-ltxr 9662  df-le 9663  df-sub 9842  df-neg 9843  df-nn 10576  df-2 10634  df-n0 10836  df-z 10905  df-uz 11127  df-fz 11725  df-fzo 11853  df-hash 12451  df-word 12589  df-s1 12592
This theorem is referenced by:  rusgranumwlkb0  25357
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