Theorem wrdl1s1 11256

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 11247	. . . 4 |- ( S e. V -> <" S "> e. Word V )
2		s1leng 11250	. . . 4 |- ( S e. V -> (♯ ` <" S "> ) = 1 )
3		s1fv 11252	. . . 4 |- ( S e. V -> ( <" S "> ` 0 ) = S )
4	1, 2, 3	3jca 1204	. . 3 |- ( S e. V -> ( <" S "> e. Word V /\ (♯ ` <" S "> ) = 1 /\ ( <" S "> ` 0 ) = S ) )
5		eleq1 2294	. . . 4 |- ( W = <" S "> -> ( W e. Word V <-> <" S "> e. Word V ) )
6		fveqeq2 5657	. . . 4 |- ( W = <" S "> -> ( (♯ ` W ) = 1 <-> (♯ ` <" S "> ) = 1 ) )
7		fveq1 5647	. . . . 5 |- ( W = <" S "> -> ( W ` 0 ) = ( <" S "> ` 0 ) )
8	7	eqeq1d 2240	. . . 4 |- ( W = <" S "> -> ( ( W ` 0 ) = S <-> ( <" S "> ` 0 ) = S ) )
9	5, 6, 8	3anbi123d 1349	. . 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 11254	. . . 4 |- ( ( W e. Word V /\ (♯ ` W ) = 1 ) -> W = <" ( W ` 0 ) "> )
12		s1eq 11245	. . . . 5 |- ( ( W ` 0 ) = S -> <" ( W ` 0 ) "> = <" S "> )
13	12	eqeq2d 2243	. . . 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 1227	. 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 1005   = wceq 1398   e. wcel 2202   ` cfv 5333   0cc0 8075   1c1 8076  ♯chash 11083  Word cword 11162   <"cs1 11241

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-addass 8177  ax-distr 8179  ax-i2m1 8180  ax-0lt1 8181  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186  ax-pre-ltirr 8187  ax-pre-ltwlin 8188  ax-pre-lttrn 8189  ax-pre-apti 8190  ax-pre-ltadd 8191

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-1o 6625  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8258  df-mnf 8259  df-xr 8260  df-ltxr 8261  df-le 8262  df-sub 8394  df-neg 8395  df-inn 9186  df-n0 9445  df-z 9524  df-uz 9800  df-fz 10289  df-fzo 10423  df-ihash 11084  df-word 11163  df-s1 11242

This theorem is referenced by: (None)