Theorem wrdl1s1 11158

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

Step	Hyp	Ref	Expression
1		s1cl 11149	. . . 4 |- ( S e. V -> <" S "> e. Word V )
2		s1leng 11152	. . . 4 |- ( S e. V -> (♯ ` <" S "> ) = 1 )
3		s1fv 11154	. . . 4 |- ( S e. V -> ( <" S "> ` 0 ) = S )
4	1, 2, 3	3jca 1201	. . 3 |- ( S e. V -> ( <" S "> e. Word V /\ (♯ ` <" S "> ) = 1 /\ ( <" S "> ` 0 ) = S ) )
5		eleq1 2292	. . . 4 |- ( W = <" S "> -> ( W e. Word V <-> <" S "> e. Word V ) )
6		fveqeq2 5635	. . . 4 |- ( W = <" S "> -> ( (♯ ` W ) = 1 <-> (♯ ` <" S "> ) = 1 ) )
7		fveq1 5625	. . . . 5 |- ( W = <" S "> -> ( W ` 0 ) = ( <" S "> ` 0 ) )
8	7	eqeq1d 2238	. . . 4 |- ( W = <" S "> -> ( ( W ` 0 ) = S <-> ( <" S "> ` 0 ) = S ) )
9	5, 6, 8	3anbi123d 1346	. . 3 |- ( W = <" S "> -> ( ( W e. Word V /\ (♯ ` W ) = 1 /\ ( W ` 0 ) = S ) <-> ( <" S "> e. Word V /\ (♯ ` <" S "> ) = 1 /\ ( <" S "> ` 0 ) = S ) ) )
10	4, 9	syl5ibrcom 157	. 2 |- ( S e. V -> ( W = <" S "> -> ( W e. Word V /\ (♯ ` W ) = 1 /\ ( W ` 0 ) = S ) ) )
11		eqs1 11156	. . . 4 |- ( ( W e. Word V /\ (♯ ` W ) = 1 ) -> W = <" ( W ` 0 ) "> )
12		s1eq 11147	. . . . 5 |- ( ( W ` 0 ) = S -> <" ( W ` 0 ) "> = <" S "> )
13	12	eqeq2d 2241	. . . 4 |- ( ( W ` 0 ) = S -> ( W = <" ( W ` 0 ) "> <-> W = <" S "> ) )
14	11, 13	syl5ibcom 155	. . 3 |- ( ( W e. Word V /\ (♯ ` W ) = 1 ) -> ( ( W ` 0 ) = S -> W = <" S "> ) )
15	14	3impia 1224	. 2 |- ( ( W e. Word V /\ (♯ ` W ) = 1 /\ ( W ` 0 ) = S ) -> W = <" S "> )
16	10, 15	impbid1 142	1 |- ( S e. V -> ( W = <" S "> <-> ( W e. Word V /\ (♯ ` W ) = 1 /\ ( W ` 0 ) = S ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   ` cfv 5317   0cc0 7995   1c1 7996  ♯chash 10992  Word cword 11066   <"cs1 11143

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 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-addass 8097  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111

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 4383  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-recs 6449  df-frec 6535  df-1o 6560  df-er 6678  df-en 6886  df-dom 6887  df-fin 6888  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-inn 9107  df-n0 9366  df-z 9443  df-uz 9719  df-fz 10201  df-fzo 10335  df-ihash 10993  df-word 11067  df-s1 11144

This theorem is referenced by: (None)