Theorem reuccatpfxs1 11238

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 2268	. . . 4 |- ( x = y -> ( x e. Word V <-> y e. Word V ) )
2		fveqeq2 5608	. . . 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 2746	. 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 2363	. . . 4 |- F/ v ( W ++ <" u "> ) e. X
7	5	nfel2 2363	. . . 4 |- F/ v ( W ++ <" x "> ) e. X
8		s1eq 11111	. . . . . 6 |- ( v = x -> <" v "> = <" x "> )
9	8	oveq2d 5983	. . . . 5 |- ( v = x -> ( W ++ <" v "> ) = ( W ++ <" x "> ) )
10	9	eleq1d 2276	. . . 4 |- ( v = x -> ( ( W ++ <" v "> ) e. X <-> ( W ++ <" x "> ) e. X ) )
11		s1eq 11111	. . . . . 6 |- ( x = u -> <" x "> = <" u "> )
12	11	oveq2d 5983	. . . . 5 |- ( x = u -> ( W ++ <" x "> ) = ( W ++ <" u "> ) )
13	12	eleq1d 2276	. . . 4 |- ( x = u -> ( ( W ++ <" x "> ) e. X <-> ( W ++ <" u "> ) e. X ) )
14	6, 7, 10, 13	reu8nf 3087	. . 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 1552	. . . . 5 |- F/ v W e. Word V
16		nfv 1552	. . . . . 6 |- F/ v ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) )
17	5, 16	nfralw 2545	. . . . 5 |- F/ v A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) )
18	15, 17	nfan 1589	. . . 4 |- F/ v ( W e. Word V /\ A. y e. X ( y e. Word V /\ (♯ ` y ) = ( (♯ ` W ) + 1 ) ) )
19		nfv 1552	. . . . 5 |- F/ v W = ( x prefix (♯ ` W ) )
20	5, 19	nfreuw 2683	. . . 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 11237	. . . . . . . . 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 1250	. . . . . . . 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 5974	. . . . . . . . . . 11 |- ( x = ( W ++ <" v "> ) -> ( x prefix (♯ ` W ) ) = ( ( W ++ <" v "> ) prefix (♯ ` W ) ) )
30		s1cl 11113	. . . . . . . . . . . . . 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 11193	. . . . . . . . . . . 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 2262	. . . . . . . . . 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 2213	. . . . . . . . 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 2581	. . . . . 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 2971	. . . . . 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 2622	. . 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 1373   e. wcel 2178   F/_wnfc 2337   A.wral 2486   E.wrex 2487   E!wreu 2488   ` cfv 5290  (class class class)co 5967   1c1 7961   + caddc 7963  ♯chash 10957  Word cword 11031   ++ cconcat 11084   <"cs1 11107   prefix cpfx 11163

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-1o 6525  df-er 6643  df-en 6851  df-dom 6852  df-fin 6853  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-fz 10166  df-fzo 10300  df-ihash 10958  df-word 11032  df-lsw 11076  df-concat 11085  df-s1 11108  df-substr 11137  df-pfx 11164

This theorem is referenced by:  reuccatpfxs1v  11239