Theorem wrdl1s1 11376

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 11367	. . . 4 |- ( S e. V -> <" S "> e. Word V )
2		s1leng 11370	. . . 4 |- ( S e. V -> (♯ ` <" S "> ) = 1 )
3		s1fv 11372	. . . 4 |- ( S e. V -> ( <" S "> ` 0 ) = S )
4	1, 2, 3	3jca 1208	. . 3 |- ( S e. V -> ( <" S "> e. Word V /\ (♯ ` <" S "> ) = 1 /\ ( <" S "> ` 0 ) = S ) )
5		eleq1 2301	. . . 4 |- ( W = <" S "> -> ( W e. Word V <-> <" S "> e. Word V ) )
6		fveqeq2 5699	. . . 4 |- ( W = <" S "> -> ( (♯ ` W ) = 1 <-> (♯ ` <" S "> ) = 1 ) )
7		fveq1 5689	. . . . 5 |- ( W = <" S "> -> ( W ` 0 ) = ( <" S "> ` 0 ) )
8	7	eqeq1d 2247	. . . 4 |- ( W = <" S "> -> ( ( W ` 0 ) = S <-> ( <" S "> ` 0 ) = S ) )
9	5, 6, 8	3anbi123d 1353	. . 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 11374	. . . 4 |- ( ( W e. Word V /\ (♯ ` W ) = 1 ) -> W = <" ( W ` 0 ) "> )
12		s1eq 11365	. . . . 5 |- ( ( W ` 0 ) = S -> <" ( W ` 0 ) "> = <" S "> )
13	12	eqeq2d 2250	. . . 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 1231	. 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 1009   = wceq 1402   e. wcel 2209   ` cfv 5372   0cc0 8169   1c1 8170  ♯chash 11192  Word cword 11282   <"cs1 11361

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-frec 6652  df-1o 6677  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-fzo 10528  df-ihash 11193  df-word 11283  df-s1 11362

This theorem is referenced by: (None)