Theorem reuccats1 13480

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

Hypothesis

Ref	Expression
reuccats1.1	|- F/_ v X

Assertion

Ref	Expression
reuccats1	|- ( ( 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 substr <. 0 , ( # ` W ) >. ) ) )

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

Allowed substitution hint:   X( v)

Proof of Theorem reuccats1

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

Step	Hyp	Ref	Expression
1		eleq1w 2684	. . . 4 |- ( x = y -> ( x e. Word V <-> y e. Word V ) )
2		fveq2 6191	. . . . 5 |- ( x = y -> ( # ` x ) = ( # ` y ) )
3	2	eqeq1d 2624	. . . 4 |- ( x = y -> ( ( # ` x ) = ( ( # ` W ) + 1 ) <-> ( # ` y ) = ( ( # ` W ) + 1 ) ) )
4	1, 3	anbi12d 747	. . 3 |- ( x = y -> ( ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) <-> ( y e. Word V /\ ( # ` y ) = ( ( # ` W ) + 1 ) ) ) )
5	4	cbvralv 3171	. 2 |- ( A. x e. X ( x e. Word V /\ ( # ` x ) = ( ( # ` W ) + 1 ) ) <-> A. y e. X ( y e. Word V /\ ( # ` y ) = ( ( # ` W ) + 1 ) ) )
6		reuccats1.1	. . . . 5 |- F/_ v X
7	6	nfel2 2781	. . . 4 |- F/ v ( W ++ <" u "> ) e. X
8	6	nfel2 2781	. . . 4 |- F/ v ( W ++ <" x "> ) e. X
9		s1eq 13380	. . . . . 6 |- ( v = x -> <" v "> = <" x "> )
10	9	oveq2d 6666	. . . . 5 |- ( v = x -> ( W ++ <" v "> ) = ( W ++ <" x "> ) )
11	10	eleq1d 2686	. . . 4 |- ( v = x -> ( ( W ++ <" v "> ) e. X <-> ( W ++ <" x "> ) e. X ) )
12		s1eq 13380	. . . . . 6 |- ( x = u -> <" x "> = <" u "> )
13	12	oveq2d 6666	. . . . 5 |- ( x = u -> ( W ++ <" x "> ) = ( W ++ <" u "> ) )
14	13	eleq1d 2686	. . . 4 |- ( x = u -> ( ( W ++ <" x "> ) e. X <-> ( W ++ <" u "> ) e. X ) )
15	7, 8, 11, 14	reu8nf 3516	. . 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 ) ) )
16		nfv 1843	. . . . 5 |- F/ v W e. Word V
17		nfv 1843	. . . . . 6 |- F/ v ( y e. Word V /\ ( # ` y ) = ( ( # ` W ) + 1 ) )
18	6, 17	nfral 2945	. . . . 5 |- F/ v A. y e. X ( y e. Word V /\ ( # ` y ) = ( ( # ` W ) + 1 ) )
19	16, 18	nfan 1828	. . . 4 |- F/ v ( W e. Word V /\ A. y e. X ( y e. Word V /\ ( # ` y ) = ( ( # ` W ) + 1 ) ) )
20		nfv 1843	. . . . 5 |- F/ v W = ( x substr <. 0 , ( # ` W ) >. )
21	6, 20	nfreu 3114	. . . 4 |- F/ v E! x e. X W = ( x substr <. 0 , ( # ` W ) >. )
22		simprl 794	. . . . . 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 )
23		simp-4l 806	. . . . . . . . 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 )
24		simpr 477	. . . . . . . . 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 e. X )
25	22	adantr 481	. . . . . . . . 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 ++ <" v "> ) e. X )
26		simplrr 801	. . . . . . . . 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 ) )
27		simp-4r 807	. . . . . . . . 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 ) ) )
28		reuccats1lem 13479	. . . . . . . . 9 |- ( ( ( W e. Word V /\ x e. X /\ ( W ++ <" v "> ) 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 substr <. 0 , ( # ` W ) >. ) -> x = ( W ++ <" v "> ) ) )
29	23, 24, 25, 26, 27, 28	syl32anc 1334	. . . . . . . 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 substr <. 0 , ( # ` W ) >. ) -> x = ( W ++ <" v "> ) ) )
30		oveq1 6657	. . . . . . . . . . 11 |- ( x = ( W ++ <" v "> ) -> ( x substr <. 0 , ( # ` W ) >. ) = ( ( W ++ <" v "> ) substr <. 0 , ( # ` W ) >. ) )
31		simpl 473	. . . . . . . . . . . . . . 15 |- ( ( W e. Word V /\ A. y e. X ( y e. Word V /\ ( # ` y ) = ( ( # ` W ) + 1 ) ) ) -> W e. Word V )
32		s1cl 13382	. . . . . . . . . . . . . . 15 |- ( v e. V -> <" v "> e. Word V )
33	31, 32	anim12i 590	. . . . . . . . . . . . . 14 |- ( ( ( 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 ) )
34	33	adantr 481	. . . . . . . . . . . . 13 |- ( ( ( ( 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 /\ <" v "> e. Word V ) )
35	34	adantr 481	. . . . . . . . . . . 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 ) )
36		swrdccat1 13457	. . . . . . . . . . . 12 |- ( ( W e. Word V /\ <" v "> e. Word V ) -> ( ( W ++ <" v "> ) substr <. 0 , ( # ` W ) >. ) = W )
37	35, 36	syl 17	. . . . . . . . . . 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 "> ) substr <. 0 , ( # ` W ) >. ) = W )
38	30, 37	sylan9eqr 2678	. . . . . . . . . 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 substr <. 0 , ( # ` W ) >. ) = W )
39	38	eqcomd 2628	. . . . . . . . 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 substr <. 0 , ( # ` W ) >. ) )
40	39	ex 450	. . . . . . . 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 substr <. 0 , ( # ` W ) >. ) ) )
41	29, 40	impbid 202	. . . . . . 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 substr <. 0 , ( # ` W ) >. ) <-> x = ( W ++ <" v "> ) ) )
42	41	ralrimiva 2966	. . . . . 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 substr <. 0 , ( # ` W ) >. ) <-> x = ( W ++ <" v "> ) ) )
43		reu6i 3397	. . . . . 6 |- ( ( ( W ++ <" v "> ) e. X /\ A. x e. X ( W = ( x substr <. 0 , ( # ` W ) >. ) <-> x = ( W ++ <" v "> ) ) ) -> E! x e. X W = ( x substr <. 0 , ( # ` W ) >. ) )
44	22, 42, 43	syl2anc 693	. . . . 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 substr <. 0 , ( # ` W ) >. ) )
45	44	exp31 630	. . . 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 substr <. 0 , ( # ` W ) >. ) ) ) )
46	19, 21, 45	rexlimd 3026	. . 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 substr <. 0 , ( # ` W ) >. ) ) )
47	15, 46	syl5bi 232	. 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 substr <. 0 , ( # ` W ) >. ) ) )
48	5, 47	sylan2b 492	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 substr <. 0 , ( # ` W ) >. ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   F/_wnfc 2751   A.wral 2912   E.wrex 2913   E!wreu 2914   <.cop 4183   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   + caddc 9939   #chash 13117  Word cword 13291   ++ cconcat 13293   <"cs1 13294   substr csubstr 13295

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-xnn0 11364  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303

This theorem is referenced by:  reuccats1v  13481  numclwlk2lem2f1o  27238