Theorem reuccatpfxs1 11318

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 2290	. . . 4 |- ( x = y -> ( x e. Word V <-> y e. Word V ) )
2		fveqeq2 5644	. . . 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 2769	. 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 2385	. . . 4 |- F/ v ( W ++ <" u "> ) e. X
7	5	nfel2 2385	. . . 4 |- F/ v ( W ++ <" x "> ) e. X
8		s1eq 11186	. . . . . 6 |- ( v = x -> <" v "> = <" x "> )
9	8	oveq2d 6029	. . . . 5 |- ( v = x -> ( W ++ <" v "> ) = ( W ++ <" x "> ) )
10	9	eleq1d 2298	. . . 4 |- ( v = x -> ( ( W ++ <" v "> ) e. X <-> ( W ++ <" x "> ) e. X ) )
11		s1eq 11186	. . . . . 6 |- ( x = u -> <" x "> = <" u "> )
12	11	oveq2d 6029	. . . . 5 |- ( x = u -> ( W ++ <" x "> ) = ( W ++ <" u "> ) )
13	12	eleq1d 2298	. . . 4 |- ( x = u -> ( ( W ++ <" x "> ) e. X <-> ( W ++ <" u "> ) e. X ) )
14	6, 7, 10, 13	reu8nf 3111	. . 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 1574	. . . . 5 |- F/ v W e. Word V
16		nfv 1574	. . . . . 6 |- F/ v ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) )
17	5, 16	nfralw 2567	. . . . 5 |- F/ v A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) )
18	15, 17	nfan 1611	. . . 4 |- F/ v ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) )
19		nfv 1574	. . . . 5 |- F/ v W = ( x prefix (♯ ` W ) )
20	5, 19	nfreuw 2706	. . . 4 |- F/ v E! x e. X W = ( x prefix (♯ ` W ) )
21		simprl 529	. . . . . 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 536	. . . . . . . . 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 542	. . . . . . . . 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 11317	. . . . . . . . 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 1271	. . . . . . . 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 6020	. . . . . . . . . . 11 |- ( x = ( W ++ <" v "> ) -> ( x prefix (♯ ` W ) ) = ( ( W ++ <" v "> ) prefix (♯ ` W ) ) )
30		s1cl 11188	. . . . . . . . . . . . . 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 11273	. . . . . . . . . . . 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 2284	. . . . . . . . . 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 2235	. . . . . . . . 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 2603	. . . . . 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 2995	. . . . . 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 2645	. . 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 1395   e. wcel 2200   F/_wnfc 2359   A.wral 2508   E.wrex 2509   E!wreu 2510   ` cfv 5324  (class class class)co 6013   1c1 8023   + caddc 8025  ♯chash 11027  Word cword 11103   ++ cconcat 11157   <"cs1 11182   prefix cpfx 11243

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 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8113  ax-resscn 8114  ax-1cn 8115  ax-1re 8116  ax-icn 8117  ax-addcl 8118  ax-addrcl 8119  ax-mulcl 8120  ax-mulrcl 8121  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-i2m1 8127  ax-0lt1 8128  ax-1rid 8129  ax-0id 8130  ax-rnegex 8131  ax-precex 8132  ax-cnre 8133  ax-pre-ltirr 8134  ax-pre-ltwlin 8135  ax-pre-lttrn 8136  ax-pre-apti 8137  ax-pre-ltadd 8138  ax-pre-mulgt0 8139

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-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-1o 6577  df-er 6697  df-en 6905  df-dom 6906  df-fin 6907  df-pnf 8206  df-mnf 8207  df-xr 8208  df-ltxr 8209  df-le 8210  df-sub 8342  df-neg 8343  df-reap 8745  df-ap 8752  df-inn 9134  df-n0 9393  df-z 9470  df-uz 9746  df-fz 10234  df-fzo 10368  df-ihash 11028  df-word 11104  df-lsw 11149  df-concat 11158  df-s1 11183  df-substr 11217  df-pfx 11244

This theorem is referenced by:  reuccatpfxs1v  11319