Theorem reuccatpfxs1 11439

Description: There is a unique word having the length of a given word increased by 1 with the given word as prefix if there is a unique symbol which extends the given word. (Contributed by Alexander van der Vekens, 6-Oct-2018.) (Revised by AV, 21-Jan-2022.) (Revised by AV, 13-Oct-2022.)

Hypothesis

Ref	Expression
reuccatpfxs1.1	|- F/_ v X

Assertion

Ref	Expression
reuccatpfxs1	|- ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ (♯ ` x ) = ( (♯ ` W ) + 1 ) ) ) -> ( E! v e. V ( W ++ <" v "> ) e. X -> E! x e. X W = ( x prefix (♯ ` W ) ) ) )

Distinct variable groups:   v, V, x   v, W, x   x, X

Allowed substitution hint:   X( v)

Proof of Theorem reuccatpfxs1

Dummy variables u y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1w 2293	. . . 4 |- ( x = y -> ( x e. Word V <-> y e. Word V ) )
2		fveqeq2 5679	. . . 4 |- ( x = y -> ( (♯ ` x ) = ( (♯ ` W ) + 1 ) <-> (♯ ` y ) = ( (♯ ` W ) + 1 ) ) )
3	1, 2	anbi12d 473	. . 3 |- ( x = y -> ( ( x e. Word V /\ (♯ ` x ) = ( (♯ ` W ) + 1 ) ) <-> ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) )
4	3	cbvralvw 2782	. 2 |- ( A. x e. X ( x e. Word V /\ (♯ ` x ) = ( (♯ ` W ) + 1 ) ) <-> A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) )
5		reuccatpfxs1.1	. . . . 5 |- F/_ v X
6	5	nfel2 2397	. . . 4 |- F/ v ( W ++ <" u "> ) e. X
7	5	nfel2 2397	. . . 4 |- F/ v ( W ++ <" x "> ) e. X
8		s1eq 11307	. . . . . 6 |- ( v = x -> <" v "> = <" x "> )
9	8	oveq2d 6066	. . . . 5 |- ( v = x -> ( W ++ <" v "> ) = ( W ++ <" x "> ) )
10	9	eleq1d 2301	. . . 4 |- ( v = x -> ( ( W ++ <" v "> ) e. X <-> ( W ++ <" x "> ) e. X ) )
11		s1eq 11307	. . . . . 6 |- ( x = u -> <" x "> = <" u "> )
12	11	oveq2d 6066	. . . . 5 |- ( x = u -> ( W ++ <" x "> ) = ( W ++ <" u "> ) )
13	12	eleq1d 2301	. . . 4 |- ( x = u -> ( ( W ++ <" x "> ) e. X <-> ( W ++ <" u "> ) e. X ) )
14	6, 7, 10, 13	reu8nf 3124	. . 3 |- ( E! v e. V ( W ++ <" v "> ) e. X <-> E. v e. V ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) )
15		nfv 1577	. . . . 5 |- F/ v W e. Word V
16		nfv 1577	. . . . . 6 |- F/ v ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) )
17	5, 16	nfralw 2579	. . . . 5 |- F/ v A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) )
18	15, 17	nfan 1614	. . . 4 |- F/ v ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) )
19		nfv 1577	. . . . 5 |- F/ v W = ( x prefix (♯ ` W ) )
20	5, 19	nfreuw 2718	. . . 4 |- F/ v E! x e. X W = ( x prefix (♯ ` W ) )
21		simprl 531	. . . . . 6 |- ( ( ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) -> ( W ++ <" v "> ) e. X )
22		simpl 109	. . . . . . . . . . 11 |- ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) -> W e. Word V )
23	22	ad2antrr 488	. . . . . . . . . 10 |- ( ( ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) -> W e. Word V )
24	23	anim1i 340	. . . . . . . . 9 |- ( ( ( ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ x e. X ) -> ( W e. Word V /\ x e. X ) )
25		simplrr 538	. . . . . . . . 9 |- ( ( ( ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ x e. X ) -> A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) )
26		simp-4r 544	. . . . . . . . 9 |- ( ( ( ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ x e. X ) -> A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) )
27		reuccatpfxs1lem 11438	. . . . . . . . 9 |- ( ( ( W e. Word V /\ x e. X ) /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) -> ( W = ( x prefix (♯ ` W ) ) -> x = ( W ++ <" v "> ) ) )
28	24, 25, 26, 27	syl3anc 1274	. . . . . . . 8 |- ( ( ( ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ x e. X ) -> ( W = ( x prefix (♯ ` W ) ) -> x = ( W ++ <" v "> ) ) )
29		oveq1 6057	. . . . . . . . . . 11 |- ( x = ( W ++ <" v "> ) -> ( x prefix (♯ ` W ) ) = ( ( W ++ <" v "> ) prefix (♯ ` W ) ) )
30		s1cl 11309	. . . . . . . . . . . . . 14 |- ( v e. V -> <" v "> e. Word V )
31	22, 30	anim12i 338	. . . . . . . . . . . . 13 |- ( ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) /\ v e. V ) -> ( W e. Word V /\ <" v "> e. Word V ) )
32	31	ad2antrr 488	. . . . . . . . . . . 12 |- ( ( ( ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ x e. X ) -> ( W e. Word V /\ <" v "> e. Word V ) )
33		pfxccat1 11394	. . . . . . . . . . . 12 |- ( ( W e. Word V /\ <" v "> e. Word V ) -> ( ( W ++ <" v "> ) prefix (♯ ` W ) ) = W )
34	32, 33	syl 14	. . . . . . . . . . 11 |- ( ( ( ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ x e. X ) -> ( ( W ++ <" v "> ) prefix (♯ ` W ) ) = W )
35	29, 34	sylan9eqr 2287	. . . . . . . . . 10 |- ( ( ( ( ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ x e. X ) /\ x = ( W ++ <" v "> ) ) -> ( x prefix (♯ ` W ) ) = W )
36	35	eqcomd 2238	. . . . . . . . 9 |- ( ( ( ( ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ x e. X ) /\ x = ( W ++ <" v "> ) ) -> W = ( x prefix (♯ ` W ) ) )
37	36	ex 115	. . . . . . . 8 |- ( ( ( ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ x e. X ) -> ( x = ( W ++ <" v "> ) -> W = ( x prefix (♯ ` W ) ) ) )
38	28, 37	impbid 129	. . . . . . 7 |- ( ( ( ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) /\ x e. X ) -> ( W = ( x prefix (♯ ` W ) ) <-> x = ( W ++ <" v "> ) ) )
39	38	ralrimiva 2615	. . . . . 6 |- ( ( ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) -> A. x e. X ( W = ( x prefix (♯ ` W ) ) <-> x = ( W ++ <" v "> ) ) )
40		reu6i 3008	. . . . . 6 |- ( ( ( W ++ <" v "> ) e. X /\ A. x e. X ( W = ( x prefix (♯ ` W ) ) <-> x = ( W ++ <" v "> ) ) ) -> E! x e. X W = ( x prefix (♯ ` W ) ) )
41	21, 39, 40	syl2anc 411	. . . . 5 |- ( ( ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) /\ v e. V ) /\ ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) ) -> E! x e. X W = ( x prefix (♯ ` W ) ) )
42	41	exp31 364	. . . 4 |- ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) -> ( v e. V -> ( ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) -> E! x e. X W = ( x prefix (♯ ` W ) ) ) ) )
43	18, 20, 42	rexlimd 2657	. . 3 |- ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) -> ( E. v e. V ( ( W ++ <" v "> ) e. X /\ A. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) ) -> E! x e. X W = ( x prefix (♯ ` W ) ) ) )
44	14, 43	biimtrid 152	. 2 |- ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) -> ( E! v e. V ( W ++ <" v "> ) e. X -> E! x e. X W = ( x prefix (♯ ` W ) ) ) )
45	4, 44	sylan2b 287	1 |- ( ( W e. Word V /\ A. x e. X ( x e. Word V /\ (♯ ` x ) = ( (♯ ` W ) + 1 ) ) ) -> ( E! v e. V ( W ++ <" v "> ) e. X -> E! x e. X W = ( x prefix (♯ ` W ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2203   F/_wnfc 2371   A.wral 2520   E.wrex 2521   E!wreu 2522   ` cfv 5352  (class class class)co 6050   1c1 8128   + caddc 8130  ♯chash 11138  Word cword 11224   ++ cconcat 11278   <"cs1 11303   prefix cpfx 11364

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 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244

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 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-frec 6622  df-1o 6647  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-inn 9238  df-n0 9497  df-z 9578  df-uz 9854  df-fz 10343  df-fzo 10477  df-ihash 11139  df-word 11225  df-lsw 11270  df-concat 11279  df-s1 11304  df-substr 11338  df-pfx 11365

This theorem is referenced by:  reuccatpfxs1v  11440