Theorem reuccatpfxs1 11497

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 2299	. . . 4 |- ( x = y -> ( x e. Word V <-> y e. Word V ) )
2		fveqeq2 5699	. . . 4 |- ( x = y -> ( (♯ ` x ) = ( (♯ ` W ) + 1 ) <-> (♯ ` y ) = ( (♯ ` W ) + 1 ) ) )
3	1, 2	anbi12d 477	. . 3 |- ( x = y -> ( ( x e. Word V /\ (♯ ` x ) = ( (♯ ` W ) + 1 ) ) <-> ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) ) )
4	3	cbvralvw 2790	. 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 2405	. . . 4 |- F/ v ( W ++ <" u "> ) e. X
7	5	nfel2 2405	. . . 4 |- F/ v ( W ++ <" x "> ) e. X
8		s1eq 11365	. . . . . 6 |- ( v = x -> <" v "> = <" x "> )
9	8	oveq2d 6091	. . . . 5 |- ( v = x -> ( W ++ <" v "> ) = ( W ++ <" x "> ) )
10	9	eleq1d 2307	. . . 4 |- ( v = x -> ( ( W ++ <" v "> ) e. X <-> ( W ++ <" x "> ) e. X ) )
11		s1eq 11365	. . . . . 6 |- ( x = u -> <" x "> = <" u "> )
12	11	oveq2d 6091	. . . . 5 |- ( x = u -> ( W ++ <" x "> ) = ( W ++ <" u "> ) )
13	12	eleq1d 2307	. . . 4 |- ( x = u -> ( ( W ++ <" x "> ) e. X <-> ( W ++ <" u "> ) e. X ) )
14	6, 7, 10, 13	reu8nf 3133	. . 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 1581	. . . . 5 |- F/ v W e. Word V
16		nfv 1581	. . . . . 6 |- F/ v ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) )
17	5, 16	nfralw 2587	. . . . 5 |- F/ v A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) )
18	15, 17	nfan 1618	. . . 4 |- F/ v ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) )
19		nfv 1581	. . . . 5 |- F/ v W = ( x prefix (♯ ` W ) )
20	5, 19	nfreuw 2726	. . . 4 |- F/ v E! x e. X W = ( x prefix (♯ ` W ) )
21		simprl 535	. . . . . 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 492	. . . . . . . . . 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 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. u e. V ( ( W ++ <" u "> ) e. X -> v = u ) )
26		simp-4r 548	. . . . . . . . 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 11496	. . . . . . . . 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 1278	. . . . . . . 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 6082	. . . . . . . . . . 11 |- ( x = ( W ++ <" v "> ) -> ( x prefix (♯ ` W ) ) = ( ( W ++ <" v "> ) prefix (♯ ` W ) ) )
30		s1cl 11367	. . . . . . . . . . . . . 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 492	. . . . . . . . . . . 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 11452	. . . . . . . . . . . 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 2293	. . . . . . . . . 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 2244	. . . . . . . . 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 2623	. . . . . 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 3017	. . . . . 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 415	. . . . 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 2665	. . 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 1402   e. wcel 2209   F/_wnfc 2379   A.wral 2528   E.wrex 2529   E!wreu 2530   ` cfv 5372  (class class class)co 6075   1c1 8170   + caddc 8172  ♯chash 11192  Word cword 11282   ++ cconcat 11336   <"cs1 11361   prefix cpfx 11422

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-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286

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-rmo 2536  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-reap 8893  df-ap 8900  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-lsw 11328  df-concat 11337  df-s1 11362  df-substr 11396  df-pfx 11423

This theorem is referenced by:  reuccatpfxs1v  11498