Theorem wwlktovfo 13701

Description: Lemma 3 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
wwlktovfo	|- ( P e. V -> F : D -onto-> 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 wwlktovfo

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

Step	Hyp	Ref	Expression
1		wrd2f1tovbij.d	. . . 4 |- D = { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) }
2		wrd2f1tovbij.r	. . . 4 |- R = { n e. V | { P , n } e. X }
3		wrd2f1tovbij.f	. . . 4 |- F = ( t e. D |-> ( t ` 1 ) )
4	1, 2, 3	wwlktovf 13699	. . 3 |- F : D --> R
5	4	a1i 11	. 2 |- ( P e. V -> F : D --> R )
6		preq2 4269	. . . . . 6 |- ( n = p -> { P , n } = { P , p } )
7	6	eleq1d 2686	. . . . 5 |- ( n = p -> ( { P , n } e. X <-> { P , p } e. X ) )
8	7, 2	elrab2 3366	. . . 4 |- ( p e. R <-> ( p e. V /\ { P , p } e. X ) )
9		simpl 473	. . . . . . . . . . 11 |- ( ( p e. V /\ { P , p } e. X ) -> p e. V )
10	9	anim2i 593	. . . . . . . . . 10 |- ( ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) -> ( P e. V /\ p e. V ) )
11		eqidd 2623	. . . . . . . . . 10 |- ( ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) -> { <. 0 , P >. , <. 1 , p >. } = { <. 0 , P >. , <. 1 , p >. } )
12		wrdlen2i 13686	. . . . . . . . . 10 |- ( ( P e. V /\ p e. V ) -> ( { <. 0 , P >. , <. 1 , p >. } = { <. 0 , P >. , <. 1 , p >. } -> ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) ) )
13	10, 11, 12	sylc 65	. . . . . . . . 9 |- ( ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) -> ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) )
14		prex 4909	. . . . . . . . . . 11 |- { <. 0 , P >. , <. 1 , p >. } e. _V
15	14	a1i 11	. . . . . . . . . 10 |- ( ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) -> { <. 0 , P >. , <. 1 , p >. } e. _V )
16		eleq1 2689	. . . . . . . . . . . . . . . . . . . 20 |- ( { <. 0 , P >. , <. 1 , p >. } = u -> ( { <. 0 , P >. , <. 1 , p >. } e. Word V <-> u e. Word V ) )
17	16	biimpd 219	. . . . . . . . . . . . . . . . . . 19 |- ( { <. 0 , P >. , <. 1 , p >. } = u -> ( { <. 0 , P >. , <. 1 , p >. } e. Word V -> u e. Word V ) )
18	17	adantr 481	. . . . . . . . . . . . . . . . . 18 |- ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) -> ( { <. 0 , P >. , <. 1 , p >. } e. Word V -> u e. Word V ) )
19	18	com12 32	. . . . . . . . . . . . . . . . 17 |- ( { <. 0 , P >. , <. 1 , p >. } e. Word V -> ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) -> u e. Word V ) )
20	19	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) -> ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) -> u e. Word V ) )
21	20	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) -> ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) -> u e. Word V ) )
22	21	impcom 446	. . . . . . . . . . . . . 14 |- ( ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) ) -> u e. Word V )
23		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . 22 |- ( { <. 0 , P >. , <. 1 , p >. } = u -> ( # ` { <. 0 , P >. , <. 1 , p >. } ) = ( # ` u ) )
24	23	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . 21 |- ( { <. 0 , P >. , <. 1 , p >. } = u -> ( ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 <-> ( # ` u ) = 2 ) )
25	24	biimpd 219	. . . . . . . . . . . . . . . . . . . 20 |- ( { <. 0 , P >. , <. 1 , p >. } = u -> ( ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 -> ( # ` u ) = 2 ) )
26	25	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) -> ( ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 -> ( # ` u ) = 2 ) )
27	26	com12 32	. . . . . . . . . . . . . . . . . 18 |- ( ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 -> ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) -> ( # ` u ) = 2 ) )
28	27	adantl 482	. . . . . . . . . . . . . . . . 17 |- ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) -> ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) -> ( # ` u ) = 2 ) )
29	28	adantr 481	. . . . . . . . . . . . . . . 16 |- ( ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) -> ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) -> ( # ` u ) = 2 ) )
30	29	impcom 446	. . . . . . . . . . . . . . 15 |- ( ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) ) -> ( # ` u ) = 2 )
31		fveq1 6190	. . . . . . . . . . . . . . . . . . . . . 22 |- ( { <. 0 , P >. , <. 1 , p >. } = u -> ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = ( u ` 0 ) )
32	31	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . 21 |- ( { <. 0 , P >. , <. 1 , p >. } = u -> ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P <-> ( u ` 0 ) = P ) )
33	32	biimpd 219	. . . . . . . . . . . . . . . . . . . 20 |- ( { <. 0 , P >. , <. 1 , p >. } = u -> ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P -> ( u ` 0 ) = P ) )
34	33	adantr 481	. . . . . . . . . . . . . . . . . . 19 |- ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) -> ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P -> ( u ` 0 ) = P ) )
35	34	com12 32	. . . . . . . . . . . . . . . . . 18 |- ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P -> ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) -> ( u ` 0 ) = P ) )
36	35	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) -> ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) -> ( u ` 0 ) = P ) )
37	36	adantl 482	. . . . . . . . . . . . . . . 16 |- ( ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) -> ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) -> ( u ` 0 ) = P ) )
38	37	impcom 446	. . . . . . . . . . . . . . 15 |- ( ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) ) -> ( u ` 0 ) = P )
39		fveq1 6190	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( { <. 0 , P >. , <. 1 , p >. } = u -> ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = ( u ` 1 ) )
40	39	eqeq1d 2624	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( { <. 0 , P >. , <. 1 , p >. } = u -> ( ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p <-> ( u ` 1 ) = p ) )
41	32, 40	anbi12d 747	. . . . . . . . . . . . . . . . . . . . . 22 |- ( { <. 0 , P >. , <. 1 , p >. } = u -> ( ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) <-> ( ( u ` 0 ) = P /\ ( u ` 1 ) = p ) ) )
42		preq12 4270	. . . . . . . . . . . . . . . . . . . . . . . . 25 |- ( ( ( u ` 0 ) = P /\ ( u ` 1 ) = p ) -> { ( u ` 0 ) , ( u ` 1 ) } = { P , p } )
43	42	eqcomd 2628	. . . . . . . . . . . . . . . . . . . . . . . 24 |- ( ( ( u ` 0 ) = P /\ ( u ` 1 ) = p ) -> { P , p } = { ( u ` 0 ) , ( u ` 1 ) } )
44	43	eleq1d 2686	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( u ` 0 ) = P /\ ( u ` 1 ) = p ) -> ( { P , p } e. X <-> { ( u ` 0 ) , ( u ` 1 ) } e. X ) )
45	44	biimpd 219	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( u ` 0 ) = P /\ ( u ` 1 ) = p ) -> ( { P , p } e. X -> { ( u ` 0 ) , ( u ` 1 ) } e. X ) )
46	41, 45	syl6bi 243	. . . . . . . . . . . . . . . . . . . . 21 |- ( { <. 0 , P >. , <. 1 , p >. } = u -> ( ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) -> ( { P , p } e. X -> { ( u ` 0 ) , ( u ` 1 ) } e. X ) ) )
47	46	com12 32	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) -> ( { <. 0 , P >. , <. 1 , p >. } = u -> ( { P , p } e. X -> { ( u ` 0 ) , ( u ` 1 ) } e. X ) ) )
48	47	adantl 482	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) -> ( { <. 0 , P >. , <. 1 , p >. } = u -> ( { P , p } e. X -> { ( u ` 0 ) , ( u ` 1 ) } e. X ) ) )
49	48	com13 88	. . . . . . . . . . . . . . . . . 18 |- ( { P , p } e. X -> ( { <. 0 , P >. , <. 1 , p >. } = u -> ( ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) -> { ( u ` 0 ) , ( u ` 1 ) } e. X ) ) )
50	49	ad2antll 765	. . . . . . . . . . . . . . . . 17 |- ( ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) -> ( { <. 0 , P >. , <. 1 , p >. } = u -> ( ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) -> { ( u ` 0 ) , ( u ` 1 ) } e. X ) ) )
51	50	impcom 446	. . . . . . . . . . . . . . . 16 |- ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) -> ( ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) -> { ( u ` 0 ) , ( u ` 1 ) } e. X ) )
52	51	imp 445	. . . . . . . . . . . . . . 15 |- ( ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) ) -> { ( u ` 0 ) , ( u ` 1 ) } e. X )
53	30, 38, 52	3jca 1242	. . . . . . . . . . . . . 14 |- ( ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) ) -> ( ( # ` u ) = 2 /\ ( u ` 0 ) = P /\ { ( u ` 0 ) , ( u ` 1 ) } e. X ) )
54		eqcom 2629	. . . . . . . . . . . . . . . . . . . 20 |- ( ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p <-> p = ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) )
55	39	eqeq2d 2632	. . . . . . . . . . . . . . . . . . . . 21 |- ( { <. 0 , P >. , <. 1 , p >. } = u -> ( p = ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) <-> p = ( u ` 1 ) ) )
56	55	biimpd 219	. . . . . . . . . . . . . . . . . . . 20 |- ( { <. 0 , P >. , <. 1 , p >. } = u -> ( p = ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) -> p = ( u ` 1 ) ) )
57	54, 56	syl5bi 232	. . . . . . . . . . . . . . . . . . 19 |- ( { <. 0 , P >. , <. 1 , p >. } = u -> ( ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p -> p = ( u ` 1 ) ) )
58	57	com12 32	. . . . . . . . . . . . . . . . . 18 |- ( ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p -> ( { <. 0 , P >. , <. 1 , p >. } = u -> p = ( u ` 1 ) ) )
59	58	ad2antll 765	. . . . . . . . . . . . . . . . 17 |- ( ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) -> ( { <. 0 , P >. , <. 1 , p >. } = u -> p = ( u ` 1 ) ) )
60	59	com12 32	. . . . . . . . . . . . . . . 16 |- ( { <. 0 , P >. , <. 1 , p >. } = u -> ( ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) -> p = ( u ` 1 ) ) )
61	60	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) -> ( ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) -> p = ( u ` 1 ) ) )
62	61	imp 445	. . . . . . . . . . . . . 14 |- ( ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) ) -> p = ( u ` 1 ) )
63	22, 53, 62	jca31 557	. . . . . . . . . . . . 13 |- ( ( ( { <. 0 , P >. , <. 1 , p >. } = u /\ ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) ) -> ( ( u e. Word V /\ ( ( # ` u ) = 2 /\ ( u ` 0 ) = P /\ { ( u ` 0 ) , ( u ` 1 ) } e. X ) ) /\ p = ( u ` 1 ) ) )
64	63	exp31 630	. . . . . . . . . . . 12 |- ( { <. 0 , P >. , <. 1 , p >. } = u -> ( ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) -> ( ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) -> ( ( u e. Word V /\ ( ( # ` u ) = 2 /\ ( u ` 0 ) = P /\ { ( u ` 0 ) , ( u ` 1 ) } e. X ) ) /\ p = ( u ` 1 ) ) ) ) )
65	64	eqcoms 2630	. . . . . . . . . . 11 |- ( u = { <. 0 , P >. , <. 1 , p >. } -> ( ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) -> ( ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) -> ( ( u e. Word V /\ ( ( # ` u ) = 2 /\ ( u ` 0 ) = P /\ { ( u ` 0 ) , ( u ` 1 ) } e. X ) ) /\ p = ( u ` 1 ) ) ) ) )
66	65	impcom 446	. . . . . . . . . 10 |- ( ( ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) /\ u = { <. 0 , P >. , <. 1 , p >. } ) -> ( ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) -> ( ( u e. Word V /\ ( ( # ` u ) = 2 /\ ( u ` 0 ) = P /\ { ( u ` 0 ) , ( u ` 1 ) } e. X ) ) /\ p = ( u ` 1 ) ) ) )
67	15, 66	spcimedv 3292	. . . . . . . . 9 |- ( ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) -> ( ( ( { <. 0 , P >. , <. 1 , p >. } e. Word V /\ ( # ` { <. 0 , P >. , <. 1 , p >. } ) = 2 ) /\ ( ( { <. 0 , P >. , <. 1 , p >. } ` 0 ) = P /\ ( { <. 0 , P >. , <. 1 , p >. } ` 1 ) = p ) ) -> E. u ( ( u e. Word V /\ ( ( # ` u ) = 2 /\ ( u ` 0 ) = P /\ { ( u ` 0 ) , ( u ` 1 ) } e. X ) ) /\ p = ( u ` 1 ) ) ) )
68	13, 67	mpd 15	. . . . . . . 8 |- ( ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) -> E. u ( ( u e. Word V /\ ( ( # ` u ) = 2 /\ ( u ` 0 ) = P /\ { ( u ` 0 ) , ( u ` 1 ) } e. X ) ) /\ p = ( u ` 1 ) ) )
69		fveq2 6191	. . . . . . . . . . . . 13 |- ( w = u -> ( # ` w ) = ( # ` u ) )
70	69	eqeq1d 2624	. . . . . . . . . . . 12 |- ( w = u -> ( ( # ` w ) = 2 <-> ( # ` u ) = 2 ) )
71		fveq1 6190	. . . . . . . . . . . . 13 |- ( w = u -> ( w ` 0 ) = ( u ` 0 ) )
72	71	eqeq1d 2624	. . . . . . . . . . . 12 |- ( w = u -> ( ( w ` 0 ) = P <-> ( u ` 0 ) = P ) )
73		fveq1 6190	. . . . . . . . . . . . . 14 |- ( w = u -> ( w ` 1 ) = ( u ` 1 ) )
74	71, 73	preq12d 4276	. . . . . . . . . . . . 13 |- ( w = u -> { ( w ` 0 ) , ( w ` 1 ) } = { ( u ` 0 ) , ( u ` 1 ) } )
75	74	eleq1d 2686	. . . . . . . . . . . 12 |- ( w = u -> ( { ( w ` 0 ) , ( w ` 1 ) } e. X <-> { ( u ` 0 ) , ( u ` 1 ) } e. X ) )
76	70, 72, 75	3anbi123d 1399	. . . . . . . . . . 11 |- ( w = u -> ( ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) <-> ( ( # ` u ) = 2 /\ ( u ` 0 ) = P /\ { ( u ` 0 ) , ( u ` 1 ) } e. X ) ) )
77	76	elrab 3363	. . . . . . . . . 10 |- ( u e. { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) } <-> ( u e. Word V /\ ( ( # ` u ) = 2 /\ ( u ` 0 ) = P /\ { ( u ` 0 ) , ( u ` 1 ) } e. X ) ) )
78	77	anbi1i 731	. . . . . . . . 9 |- ( ( u e. { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) } /\ p = ( u ` 1 ) ) <-> ( ( u e. Word V /\ ( ( # ` u ) = 2 /\ ( u ` 0 ) = P /\ { ( u ` 0 ) , ( u ` 1 ) } e. X ) ) /\ p = ( u ` 1 ) ) )
79	78	exbii 1774	. . . . . . . 8 |- ( E. u ( u e. { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) } /\ p = ( u ` 1 ) ) <-> E. u ( ( u e. Word V /\ ( ( # ` u ) = 2 /\ ( u ` 0 ) = P /\ { ( u ` 0 ) , ( u ` 1 ) } e. X ) ) /\ p = ( u ` 1 ) ) )
80	68, 79	sylibr 224	. . . . . . 7 |- ( ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) -> E. u ( u e. { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) } /\ p = ( u ` 1 ) ) )
81		df-rex 2918	. . . . . . 7 |- ( E. u e. { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) } p = ( u ` 1 ) <-> E. u ( u e. { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) } /\ p = ( u ` 1 ) ) )
82	80, 81	sylibr 224	. . . . . 6 |- ( ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) -> E. u e. { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) } p = ( u ` 1 ) )
83	1	rexeqi 3143	. . . . . 6 |- ( E. u e. D p = ( u ` 1 ) <-> E. u e. { w e. Word V | ( ( # ` w ) = 2 /\ ( w ` 0 ) = P /\ { ( w ` 0 ) , ( w ` 1 ) } e. X ) } p = ( u ` 1 ) )
84	82, 83	sylibr 224	. . . . 5 |- ( ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) -> E. u e. D p = ( u ` 1 ) )
85		fveq1 6190	. . . . . . . 8 |- ( t = u -> ( t ` 1 ) = ( u ` 1 ) )
86		fvex 6201	. . . . . . . 8 |- ( u ` 1 ) e. _V
87	85, 3, 86	fvmpt 6282	. . . . . . 7 |- ( u e. D -> ( F ` u ) = ( u ` 1 ) )
88	87	eqeq2d 2632	. . . . . 6 |- ( u e. D -> ( p = ( F ` u ) <-> p = ( u ` 1 ) ) )
89	88	rexbiia 3040	. . . . 5 |- ( E. u e. D p = ( F ` u ) <-> E. u e. D p = ( u ` 1 ) )
90	84, 89	sylibr 224	. . . 4 |- ( ( P e. V /\ ( p e. V /\ { P , p } e. X ) ) -> E. u e. D p = ( F ` u ) )
91	8, 90	sylan2b 492	. . 3 |- ( ( P e. V /\ p e. R ) -> E. u e. D p = ( F ` u ) )
92	91	ralrimiva 2966	. 2 |- ( P e. V -> A. p e. R E. u e. D p = ( F ` u ) )
93		dffo3 6374	. 2 |- ( F : D -onto-> R <-> ( F : D --> R /\ A. p e. R E. u e. D p = ( F ` u ) ) )
94	5, 92, 93	sylanbrc 698	1 |- ( P e. V -> F : D -onto-> R )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   E.wex 1704   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   {cpr 4179   <.cop 4183   |-> cmpt 4729   -->wf 5884   -onto->wfo 5886   ` cfv 5888   0cc0 9936   1c1 9937   2c2 11070   #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-rmo 2920  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-cda 8990  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