Theorem wwlktovf1 13700

Description: Lemma 2 for wrd2f1tovbij 13703. (Contributed by Alexander van der Vekens, 27-Jul-2018.)

Hypotheses

Ref	Expression
wrd2f1tovbij.d	|- D = { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) }
wrd2f1tovbij.r	|- R = { n e. V | { P , n } e. X }
wrd2f1tovbij.f	|- F = ( t e. D |-> ( t ` 1 ) )

Assertion

Ref	Expression
wwlktovf1	|- F : D -1-1-> R

Distinct variable groups:   t, D   P, n, t, w   t, R   n, V, t, w   n, X, w

Allowed substitution hints:   D( w, n)   R( w, n)   F( w, t, n)   X( t)

Proof of Theorem wwlktovf1

Dummy variables x y i are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		wrd2f1tovbij.d	. . 3 |- D = { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) }
2		wrd2f1tovbij.r	. . 3 |- R = { n e. V | { P , n } e. X }
3		wrd2f1tovbij.f	. . 3 |- F = ( t e. D |-> ( t ` 1 ) )
4	1, 2, 3	wwlktovf 13699	. 2 |- F : D --> R
5		fveq1 6190	. . . . . 6 |- ( t = x -> ( t ` 1 ) = ( x ` 1 ) )
6		fvex 6201	. . . . . 6 |- ( x ` 1 ) e. _V
7	5, 3, 6	fvmpt 6282	. . . . 5 |- ( x e. D -> ( F ` x ) = ( x ` 1 ) )
8		fveq1 6190	. . . . . 6 |- ( t = y -> ( t ` 1 ) = ( y ` 1 ) )
9		fvex 6201	. . . . . 6 |- ( y ` 1 ) e. _V
10	8, 3, 9	fvmpt 6282	. . . . 5 |- ( y e. D -> ( F ` y ) = ( y ` 1 ) )
11	7, 10	eqeqan12d 2638	. . . 4 |- ( ( x e. D /\ y e. D ) -> ( ( F ` x ) = ( F ` y ) <-> ( x ` 1 ) = ( y ` 1 ) ) )
12		fveq2 6191	. . . . . . . 8 |- ( w = x -> ( # ` w ) = ( # ` x ) )
13	12	eqeq1d 2624	. . . . . . 7 |- ( w = x -> ( ( # ` w ) = 2 <-> ( # ` x ) = 2 ) )
14		fveq1 6190	. . . . . . . 8 |- ( w = x -> ( w ` 0 ) = ( x ` 0 ) )
15	14	eqeq1d 2624	. . . . . . 7 |- ( w = x -> ( ( w ` 0 ) = P <-> ( x ` 0 ) = P ) )
16		fveq1 6190	. . . . . . . . 9 |- ( w = x -> ( w ` 1 ) = ( x ` 1 ) )
17	14, 16	preq12d 4276	. . . . . . . 8 |- ( w = x -> { ( w ` 0 ) , ( w ` 1 ) } = { ( x ` 0 ) , ( x ` 1 ) } )
18	17	eleq1d 2686	. . . . . . 7 |- ( w = x -> ( { ( w ` 0 ) , ( w ` 1 ) } e. X <-> { ( x ` 0 ) , ( x ` 1 ) } e. X ) )
19	13, 15, 18	3anbi123d 1399	. . . . . 6 |- ( w = x -> ( ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) <-> ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) ) )
20	19, 1	elrab2 3366	. . . . 5 |- ( x e. D <-> ( x e. Word V /\ ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) ) )
21		fveq2 6191	. . . . . . . 8 |- ( w = y -> ( # ` w ) = ( # ` y ) )
22	21	eqeq1d 2624	. . . . . . 7 |- ( w = y -> ( ( # ` w ) = 2 <-> ( # ` y ) = 2 ) )
23		fveq1 6190	. . . . . . . 8 |- ( w = y -> ( w ` 0 ) = ( y ` 0 ) )
24	23	eqeq1d 2624	. . . . . . 7 |- ( w = y -> ( ( w ` 0 ) = P <-> ( y ` 0 ) = P ) )
25		fveq1 6190	. . . . . . . . 9 |- ( w = y -> ( w ` 1 ) = ( y ` 1 ) )
26	23, 25	preq12d 4276	. . . . . . . 8 |- ( w = y -> { ( w ` 0 ) , ( w ` 1 ) } = { ( y ` 0 ) , ( y ` 1 ) } )
27	26	eleq1d 2686	. . . . . . 7 |- ( w = y -> ( { ( w ` 0 ) , ( w ` 1 ) } e. X <-> { ( y ` 0 ) , ( y ` 1 ) } e. X ) )
28	22, 24, 27	3anbi123d 1399	. . . . . 6 |- ( w = y -> ( ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) <-> ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) ) )
29	28, 1	elrab2 3366	. . . . 5 |- ( y e. D <-> ( y e. Word V /\ ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) ) )
30		simp1 1061	. . . . . . . . . 10 |- ( ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) -> ( # ` x ) = 2 )
31	30	adantl 482	. . . . . . . . 9 |- ( ( x e. Word V /\ ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) ) -> ( # ` x ) = 2 )
32		simp1 1061	. . . . . . . . . . 11 |- ( ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) -> ( # ` y ) = 2 )
33	32	eqcomd 2628	. . . . . . . . . 10 |- ( ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) -> 2 = ( # ` y ) )
34	33	adantl 482	. . . . . . . . 9 |- ( ( y e. Word V /\ ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) ) -> 2 = ( # ` y ) )
35	31, 34	sylan9eq 2676	. . . . . . . 8 |- ( ( ( x e. Word V /\ ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) ) /\ ( y e. Word V /\ ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) ) ) -> ( # ` x ) = ( # ` y ) )
36	35	adantr 481	. . . . . . 7 |- ( ( ( ( x e. Word V /\ ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) ) /\ ( y e. Word V /\ ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) ) ) /\ ( x ` 1 ) = ( y ` 1 ) ) -> ( # ` x ) = ( # ` y ) )
37		simp2 1062	. . . . . . . . . . 11 |- ( ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) -> ( x ` 0 ) = P )
38	37	adantl 482	. . . . . . . . . 10 |- ( ( x e. Word V /\ ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) ) -> ( x ` 0 ) = P )
39		simp2 1062	. . . . . . . . . . . 12 |- ( ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) -> ( y ` 0 ) = P )
40	39	eqcomd 2628	. . . . . . . . . . 11 |- ( ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) -> P = ( y ` 0 ) )
41	40	adantl 482	. . . . . . . . . 10 |- ( ( y e. Word V /\ ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) ) -> P = ( y ` 0 ) )
42	38, 41	sylan9eq 2676	. . . . . . . . 9 |- ( ( ( x e. Word V /\ ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) ) /\ ( y e. Word V /\ ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) ) ) -> ( x ` 0 ) = ( y ` 0 ) )
43	42	adantr 481	. . . . . . . 8 |- ( ( ( ( x e. Word V /\ ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) ) /\ ( y e. Word V /\ ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) ) ) /\ ( x ` 1 ) = ( y ` 1 ) ) -> ( x ` 0 ) = ( y ` 0 ) )
44		simpr 477	. . . . . . . 8 |- ( ( ( ( x e. Word V /\ ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) ) /\ ( y e. Word V /\ ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) ) ) /\ ( x ` 1 ) = ( y ` 1 ) ) -> ( x ` 1 ) = ( y ` 1 ) )
45		oveq2 6658	. . . . . . . . . . . . 13 |- ( ( # ` x ) = 2 -> ( 0..^ ( # ` x ) ) = ( 0..^ 2 ) )
46		fzo0to2pr 12553	. . . . . . . . . . . . 13 |- ( 0..^ 2 ) = { 0 , 1 }
47	45, 46	syl6eq 2672	. . . . . . . . . . . 12 |- ( ( # ` x ) = 2 -> ( 0..^ ( # ` x ) ) = { 0 , 1 } )
48	47	raleqdv 3144	. . . . . . . . . . 11 |- ( ( # ` x ) = 2 -> ( A. i e. ( 0..^ ( # ` x ) ) ( x ` i ) = ( y ` i ) <-> A. i e. { 0 , 1 } ( x ` i ) = ( y ` i ) ) )
49		c0ex 10034	. . . . . . . . . . . 12 |- 0 e. _V
50		1ex 10035	. . . . . . . . . . . 12 |- 1 e. _V
51		fveq2 6191	. . . . . . . . . . . . 13 |- ( i = 0 -> ( x ` i ) = ( x ` 0 ) )
52		fveq2 6191	. . . . . . . . . . . . 13 |- ( i = 0 -> ( y ` i ) = ( y ` 0 ) )
53	51, 52	eqeq12d 2637	. . . . . . . . . . . 12 |- ( i = 0 -> ( ( x ` i ) = ( y ` i ) <-> ( x ` 0 ) = ( y ` 0 ) ) )
54		fveq2 6191	. . . . . . . . . . . . 13 |- ( i = 1 -> ( x ` i ) = ( x ` 1 ) )
55		fveq2 6191	. . . . . . . . . . . . 13 |- ( i = 1 -> ( y ` i ) = ( y ` 1 ) )
56	54, 55	eqeq12d 2637	. . . . . . . . . . . 12 |- ( i = 1 -> ( ( x ` i ) = ( y ` i ) <-> ( x ` 1 ) = ( y ` 1 ) ) )
57	49, 50, 53, 56	ralpr 4238	. . . . . . . . . . 11 |- ( A. i e. { 0 , 1 } ( x ` i ) = ( y ` i ) <-> ( ( x ` 0 ) = ( y ` 0 ) /\ ( x ` 1 ) = ( y ` 1 ) ) )
58	48, 57	syl6bb 276	. . . . . . . . . 10 |- ( ( # ` x ) = 2 -> ( A. i e. ( 0..^ ( # ` x ) ) ( x ` i ) = ( y ` i ) <-> ( ( x ` 0 ) = ( y ` 0 ) /\ ( x ` 1 ) = ( y ` 1 ) ) ) )
59	58	3ad2ant1 1082	. . . . . . . . 9 |- ( ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) -> ( A. i e. ( 0..^ ( # ` x ) ) ( x ` i ) = ( y ` i ) <-> ( ( x ` 0 ) = ( y ` 0 ) /\ ( x ` 1 ) = ( y ` 1 ) ) ) )
60	59	ad3antlr 767	. . . . . . . 8 |- ( ( ( ( x e. Word V /\ ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) ) /\ ( y e. Word V /\ ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) ) ) /\ ( x ` 1 ) = ( y ` 1 ) ) -> ( A. i e. ( 0..^ ( # ` x ) ) ( x ` i ) = ( y ` i ) <-> ( ( x ` 0 ) = ( y ` 0 ) /\ ( x ` 1 ) = ( y ` 1 ) ) ) )
61	43, 44, 60	mpbir2and 957	. . . . . . 7 |- ( ( ( ( x e. Word V /\ ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) ) /\ ( y e. Word V /\ ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) ) ) /\ ( x ` 1 ) = ( y ` 1 ) ) -> A. i e. ( 0..^ ( # ` x ) ) ( x ` i ) = ( y ` i ) )
62		simpl 473	. . . . . . . . . 10 |- ( ( x e. Word V /\ ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) ) -> x e. Word V )
63		simpl 473	. . . . . . . . . 10 |- ( ( y e. Word V /\ ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) ) -> y e. Word V )
64	62, 63	anim12i 590	. . . . . . . . 9 |- ( ( ( x e. Word V /\ ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) ) /\ ( y e. Word V /\ ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) ) ) -> ( x e. Word V /\ y e. Word V ) )
65	64	adantr 481	. . . . . . . 8 |- ( ( ( ( x e. Word V /\ ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) ) /\ ( y e. Word V /\ ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) ) ) /\ ( x ` 1 ) = ( y ` 1 ) ) -> ( x e. Word V /\ y e. Word V ) )
66		eqwrd 13346	. . . . . . . 8 |- ( ( x e. Word V /\ y e. Word V ) -> ( x = y <-> ( ( # ` x ) = ( # ` y ) /\ A. i e. ( 0..^ ( # ` x ) ) ( x ` i ) = ( y ` i ) ) ) )
67	65, 66	syl 17	. . . . . . 7 |- ( ( ( ( x e. Word V /\ ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) ) /\ ( y e. Word V /\ ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) ) ) /\ ( x ` 1 ) = ( y ` 1 ) ) -> ( x = y <-> ( ( # ` x ) = ( # ` y ) /\ A. i e. ( 0..^ ( # ` x ) ) ( x ` i ) = ( y ` i ) ) ) )
68	36, 61, 67	mpbir2and 957	. . . . . 6 |- ( ( ( ( x e. Word V /\ ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) ) /\ ( y e. Word V /\ ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) ) ) /\ ( x ` 1 ) = ( y ` 1 ) ) -> x = y )
69	68	ex 450	. . . . 5 |- ( ( ( x e. Word V /\ ( ( # ` x ) = 2 /\ ( x ` 0 ) = P /\ { ( x ` 0 ) , ( x ` 1 ) } e. X ) ) /\ ( y e. Word V /\ ( ( # ` y ) = 2 /\ ( y ` 0 ) = P /\ { ( y ` 0 ) , ( y ` 1 ) } e. X ) ) ) -> ( ( x ` 1 ) = ( y ` 1 ) -> x = y ) )
70	20, 29, 69	syl2anb 496	. . . 4 |- ( ( x e. D /\ y e. D ) -> ( ( x ` 1 ) = ( y ` 1 ) -> x = y ) )
71	11, 70	sylbid 230	. . 3 |- ( ( x e. D /\ y e. D ) -> ( ( F ` x ) = ( F ` y ) -> x = y ) )
72	71	rgen2a 2977	. 2 |- A. x e. D A. y e. D ( ( F ` x ) = ( F ` y ) -> x = y )
73		dff13 6512	. 2 |- ( F : D -1-1-> R <-> ( F : D --> R /\ A. x e. D A. y e. D ( ( F ` x ) = ( F ` y ) -> x = y ) ) )
74	4, 72, 73	mpbir2an 955	1 |- F : D -1-1-> R

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   {cpr 4179   |-> cmpt 4729   -->wf 5884   -1-1->wf1 5885   ` cfv 5888  (class class class)co 6650   0cc0 9936   1c1 9937   2c2 11070  ..^cfzo 12465   #chash 13117  Word cword 13291

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-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299

This theorem is referenced by:  wwlktovf1o  13702