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Theorem wrdl1s1 12579
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 12571 . . . 4  |-  ( S  e.  V  ->  <" S ">  e. Word  V )
2 s1len 12574 . . . . 5  |-  ( # `  <" S "> )  =  1
32a1i 11 . . . 4  |-  ( S  e.  V  ->  ( # `
 <" S "> )  =  1
)
4 s1fv 12576 . . . 4  |-  ( S  e.  V  ->  ( <" S "> `  0 )  =  S )
51, 3, 43jca 1176 . . 3  |-  ( S  e.  V  ->  ( <" S ">  e. Word  V  /\  ( # `  <" S "> )  =  1  /\  ( <" S "> `  0 )  =  S ) )
6 eleq1 2539 . . . 4  |-  ( W  =  <" S ">  ->  ( W  e. Word  V 
<-> 
<" S ">  e. Word  V ) )
7 fveq2 5864 . . . . 5  |-  ( W  =  <" S ">  ->  ( # `  W
)  =  ( # `  <" S "> ) )
87eqeq1d 2469 . . . 4  |-  ( W  =  <" S ">  ->  ( ( # `  W )  =  1  <-> 
( # `  <" S "> )  =  1 ) )
9 fveq1 5863 . . . . 5  |-  ( W  =  <" S ">  ->  ( W ` 
0 )  =  (
<" S "> `  0 ) )
109eqeq1d 2469 . . . 4  |-  ( W  =  <" S ">  ->  ( ( W `
 0 )  =  S  <->  ( <" S "> `  0 )  =  S ) )
116, 8, 103anbi123d 1299 . . 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 12578 . . . 4  |-  ( ( W  e. Word  V  /\  ( # `  W )  =  1 )  ->  W  =  <" ( W `  0 ) "> )
14 s1eq 12569 . . . . 5  |-  ( ( W `  0 )  =  S  ->  <" ( W `  0 ) ">  =  <" S "> )
1514eqeq2d 2481 . . . 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 1193 . 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ` cfv 5586   0cc0 9488   1c1 9489   #chash 12367  Word cword 12494   <"cs1 12497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-card 8316  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-fz 11669  df-fzo 11789  df-hash 12368  df-word 12502  df-s1 12505
This theorem is referenced by:  rusgranumwlkb0  24626
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