Theorem efgsfo 18152

Description: For any word, there is a sequence of extensions starting at a reduced word and ending at the target word, such that each word in the chain is an extension of the previous (inserting an element and its inverse at adjacent indexes somewhere in the sequence). (Contributed by Mario Carneiro, 27-Sep-2015.)

Hypotheses

Ref	Expression
efgval.w	|- W = ( _I ` Word ( I X. 2o ) )
efgval.r	|- .~ = ( ~FG ` I )
efgval2.m	|- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
efgval2.t	|- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
efgred.d	|- D = ( W \ U_ x e. W ran ( T ` x ) )
efgred.s	|- S = ( m e. { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) )

Assertion

Ref	Expression
efgsfo	|- S : dom S -onto-> W

Distinct variable groups:   y, z   t, n, v, w, y, z, m, x   m, M   x, n, M, t, v, w   k, m, t, x, T   k, n, v, w, y, z, W, m, t, x   .~ , m, t, x, y, z   m, I, n, t, v, w, x, y, z   D, m, t

Allowed substitution hints:   D( x, y, z, w, v, k, n)   .~ ( w, v, k, n)   S( x, y, z, w, v, t, k, m, n)   T( y, z, w, v, n)   I( k)   M( y, z, k)

Proof of Theorem efgsfo

Dummy variables a b c d i o are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		efgval.w	. . . 4 |- W = ( _I ` Word ( I X. 2o ) )
2		efgval.r	. . . 4 |- .~ = ( ~FG ` I )
3		efgval2.m	. . . 4 |- M = ( y e. I , z e. 2o |-> <. y , ( 1o \ z ) >. )
4		efgval2.t	. . . 4 |- T = ( v e. W |-> ( n e. ( 0 ... ( # ` v ) ) , w e. ( I X. 2o ) |-> ( v splice <. n , n , <" w ( M ` w ) "> >. ) ) )
5		efgred.d	. . . 4 |- D = ( W \ U_ x e. W ran ( T ` x ) )
6		efgred.s	. . . 4 |- S = ( m e. { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } |-> ( m ` ( ( # ` m ) - 1 ) ) )
7	1, 2, 3, 4, 5, 6	efgsf 18142	. . 3 |- S : { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } --> W
8	7	fdmi 6052	. . . 4 |- dom S = { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) }
9	8	feq2i 6037	. . 3 |- ( S : dom S --> W <-> S : { t e. (Word W \ { (/) } ) | ( ( t ` 0 ) e. D /\ A. k e. ( 1..^ ( # ` t ) ) ( t ` k ) e. ran ( T ` ( t ` ( k - 1 ) ) ) ) } --> W )
10	7, 9	mpbir 221	. 2 |- S : dom S --> W
11		frn 6053	. . . 4 |- ( S : dom S --> W -> ran S C_ W )
12	10, 11	ax-mp 5	. . 3 |- ran S C_ W
13		fviss 6256	. . . . . . . . 9 |- ( _I ` Word ( I X. 2o ) ) C_ Word ( I X. 2o )
14	1, 13	eqsstri 3635	. . . . . . . 8 |- W C_ Word ( I X. 2o )
15	14	sseli 3599	. . . . . . 7 |- ( c e. W -> c e. Word ( I X. 2o ) )
16		lencl 13324	. . . . . . 7 |- ( c e. Word ( I X. 2o ) -> ( # ` c ) e. NN0 )
17	15, 16	syl 17	. . . . . 6 |- ( c e. W -> ( # ` c ) e. NN0 )
18		peano2nn0 11333	. . . . . 6 |- ( ( # ` c ) e. NN0 -> ( ( # ` c ) + 1 ) e. NN0 )
19	14	sseli 3599	. . . . . . . . . . . 12 |- ( a e. W -> a e. Word ( I X. 2o ) )
20		lencl 13324	. . . . . . . . . . . 12 |- ( a e. Word ( I X. 2o ) -> ( # ` a ) e. NN0 )
21	19, 20	syl 17	. . . . . . . . . . 11 |- ( a e. W -> ( # ` a ) e. NN0 )
22		nn0nlt0 11319	. . . . . . . . . . . 12 |- ( ( # ` a ) e. NN0 -> -. ( # ` a ) < 0 )
23		breq2 4657	. . . . . . . . . . . . 13 |- ( b = 0 -> ( ( # ` a ) < b <-> ( # ` a ) < 0 ) )
24	23	notbid 308	. . . . . . . . . . . 12 |- ( b = 0 -> ( -. ( # ` a ) < b <-> -. ( # ` a ) < 0 ) )
25	22, 24	syl5ibr 236	. . . . . . . . . . 11 |- ( b = 0 -> ( ( # ` a ) e. NN0 -> -. ( # ` a ) < b ) )
26	21, 25	syl5 34	. . . . . . . . . 10 |- ( b = 0 -> ( a e. W -> -. ( # ` a ) < b ) )
27	26	ralrimiv 2965	. . . . . . . . 9 |- ( b = 0 -> A. a e. W -. ( # ` a ) < b )
28		rabeq0 3957	. . . . . . . . 9 |- ( { a e. W | ( # ` a ) < b } = (/) <-> A. a e. W -. ( # ` a ) < b )
29	27, 28	sylibr 224	. . . . . . . 8 |- ( b = 0 -> { a e. W | ( # ` a ) < b } = (/) )
30	29	sseq1d 3632	. . . . . . 7 |- ( b = 0 -> ( { a e. W | ( # ` a ) < b } C_ ran S <-> (/) C_ ran S ) )
31		breq2 4657	. . . . . . . . 9 |- ( b = d -> ( ( # ` a ) < b <-> ( # ` a ) < d ) )
32	31	rabbidv 3189	. . . . . . . 8 |- ( b = d -> { a e. W | ( # ` a ) < b } = { a e. W | ( # ` a ) < d } )
33	32	sseq1d 3632	. . . . . . 7 |- ( b = d -> ( { a e. W | ( # ` a ) < b } C_ ran S <-> { a e. W | ( # ` a ) < d } C_ ran S ) )
34		breq2 4657	. . . . . . . . 9 |- ( b = ( d + 1 ) -> ( ( # ` a ) < b <-> ( # ` a ) < ( d + 1 ) ) )
35	34	rabbidv 3189	. . . . . . . 8 |- ( b = ( d + 1 ) -> { a e. W | ( # ` a ) < b } = { a e. W | ( # ` a ) < ( d + 1 ) } )
36	35	sseq1d 3632	. . . . . . 7 |- ( b = ( d + 1 ) -> ( { a e. W | ( # ` a ) < b } C_ ran S <-> { a e. W | ( # ` a ) < ( d + 1 ) } C_ ran S ) )
37		breq2 4657	. . . . . . . . 9 |- ( b = ( ( # ` c ) + 1 ) -> ( ( # ` a ) < b <-> ( # ` a ) < ( ( # ` c ) + 1 ) ) )
38	37	rabbidv 3189	. . . . . . . 8 |- ( b = ( ( # ` c ) + 1 ) -> { a e. W | ( # ` a ) < b } = { a e. W | ( # ` a ) < ( ( # ` c ) + 1 ) } )
39	38	sseq1d 3632	. . . . . . 7 |- ( b = ( ( # ` c ) + 1 ) -> ( { a e. W | ( # ` a ) < b } C_ ran S <-> { a e. W | ( # ` a ) < ( ( # ` c ) + 1 ) } C_ ran S ) )
40		0ss 3972	. . . . . . 7 |- (/) C_ ran S
41		simpr 477	. . . . . . . . . 10 |- ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) -> { a e. W | ( # ` a ) < d } C_ ran S )
42		fveq2 6191	. . . . . . . . . . . . 13 |- ( a = c -> ( # ` a ) = ( # ` c ) )
43	42	eqeq1d 2624	. . . . . . . . . . . 12 |- ( a = c -> ( ( # ` a ) = d <-> ( # ` c ) = d ) )
44	43	cbvrabv 3199	. . . . . . . . . . 11 |- { a e. W | ( # ` a ) = d } = { c e. W | ( # ` c ) = d }
45		eliun 4524	. . . . . . . . . . . . . . 15 |- ( c e. U_ x e. W ran ( T ` x ) <-> E. x e. W c e. ran ( T ` x ) )
46		fveq2 6191	. . . . . . . . . . . . . . . . . 18 |- ( x = b -> ( T ` x ) = ( T ` b ) )
47	46	rneqd 5353	. . . . . . . . . . . . . . . . 17 |- ( x = b -> ran ( T ` x ) = ran ( T ` b ) )
48	47	eleq2d 2687	. . . . . . . . . . . . . . . 16 |- ( x = b -> ( c e. ran ( T ` x ) <-> c e. ran ( T ` b ) ) )
49	48	cbvrexv 3172	. . . . . . . . . . . . . . 15 |- ( E. x e. W c e. ran ( T ` x ) <-> E. b e. W c e. ran ( T ` b ) )
50	45, 49	bitri 264	. . . . . . . . . . . . . 14 |- ( c e. U_ x e. W ran ( T ` x ) <-> E. b e. W c e. ran ( T ` b ) )
51		simpl1r 1113	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( b e. W /\ c e. ran ( T ` b ) ) ) -> { a e. W | ( # ` a ) < d } C_ ran S )
52		simprl 794	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( b e. W /\ c e. ran ( T ` b ) ) ) -> b e. W )
53	14, 52	sseldi 3601	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( b e. W /\ c e. ran ( T ` b ) ) ) -> b e. Word ( I X. 2o ) )
54		lencl 13324	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( b e. Word ( I X. 2o ) -> ( # ` b ) e. NN0 )
55	53, 54	syl 17	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( b e. W /\ c e. ran ( T ` b ) ) ) -> ( # ` b ) e. NN0 )
56	55	nn0red 11352	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( b e. W /\ c e. ran ( T ` b ) ) ) -> ( # ` b ) e. RR )
57		2rp 11837	. . . . . . . . . . . . . . . . . . . . 21 |- 2 e. RR+
58		ltaddrp 11867	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( # ` b ) e. RR /\ 2 e. RR+ ) -> ( # ` b ) < ( ( # ` b ) + 2 ) )
59	56, 57, 58	sylancl 694	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( b e. W /\ c e. ran ( T ` b ) ) ) -> ( # ` b ) < ( ( # ` b ) + 2 ) )
60	1, 2, 3, 4	efgtlen 18139	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( b e. W /\ c e. ran ( T ` b ) ) -> ( # ` c ) = ( ( # ` b ) + 2 ) )
61	60	adantl 482	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( b e. W /\ c e. ran ( T ` b ) ) ) -> ( # ` c ) = ( ( # ` b ) + 2 ) )
62		simpl3 1066	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( b e. W /\ c e. ran ( T ` b ) ) ) -> ( # ` c ) = d )
63	61, 62	eqtr3d 2658	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( b e. W /\ c e. ran ( T ` b ) ) ) -> ( ( # ` b ) + 2 ) = d )
64	59, 63	breqtrd 4679	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( b e. W /\ c e. ran ( T ` b ) ) ) -> ( # ` b ) < d )
65		fveq2 6191	. . . . . . . . . . . . . . . . . . . . 21 |- ( a = b -> ( # ` a ) = ( # ` b ) )
66	65	breq1d 4663	. . . . . . . . . . . . . . . . . . . 20 |- ( a = b -> ( ( # ` a ) < d <-> ( # ` b ) < d ) )
67	66	elrab 3363	. . . . . . . . . . . . . . . . . . 19 |- ( b e. { a e. W | ( # ` a ) < d } <-> ( b e. W /\ ( # ` b ) < d ) )
68	52, 64, 67	sylanbrc 698	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( b e. W /\ c e. ran ( T ` b ) ) ) -> b e. { a e. W | ( # ` a ) < d } )
69	51, 68	sseldd 3604	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( b e. W /\ c e. ran ( T ` b ) ) ) -> b e. ran S )
70		ffn 6045	. . . . . . . . . . . . . . . . . . 19 |- ( S : dom S --> W -> S Fn dom S )
71	10, 70	ax-mp 5	. . . . . . . . . . . . . . . . . 18 |- S Fn dom S
72		fvelrnb 6243	. . . . . . . . . . . . . . . . . 18 |- ( S Fn dom S -> ( b e. ran S <-> E. o e. dom S ( S ` o ) = b ) )
73	71, 72	ax-mp 5	. . . . . . . . . . . . . . . . 17 |- ( b e. ran S <-> E. o e. dom S ( S ` o ) = b )
74	69, 73	sylib 208	. . . . . . . . . . . . . . . 16 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( b e. W /\ c e. ran ( T ` b ) ) ) -> E. o e. dom S ( S ` o ) = b )
75		simprrl 804	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( ( b e. W /\ c e. ran ( T ` b ) ) /\ ( o e. dom S /\ ( S ` o ) = b ) ) ) -> o e. dom S )
76	1, 2, 3, 4, 5, 6	efgsdm 18143	. . . . . . . . . . . . . . . . . . . . 21 |- ( o e. dom S <-> ( o e. (Word W \ { (/) } ) /\ ( o ` 0 ) e. D /\ A. i e. ( 1..^ ( # ` o ) ) ( o ` i ) e. ran ( T ` ( o ` ( i - 1 ) ) ) ) )
77	76	simp1bi 1076	. . . . . . . . . . . . . . . . . . . 20 |- ( o e. dom S -> o e. (Word W \ { (/) } ) )
78		eldifi 3732	. . . . . . . . . . . . . . . . . . . 20 |- ( o e. (Word W \ { (/) } ) -> o e. Word W )
79	75, 77, 78	3syl 18	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( ( b e. W /\ c e. ran ( T ` b ) ) /\ ( o e. dom S /\ ( S ` o ) = b ) ) ) -> o e. Word W )
80		simpl2 1065	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( ( b e. W /\ c e. ran ( T ` b ) ) /\ ( o e. dom S /\ ( S ` o ) = b ) ) ) -> c e. W )
81		simprlr 803	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( ( b e. W /\ c e. ran ( T ` b ) ) /\ ( o e. dom S /\ ( S ` o ) = b ) ) ) -> c e. ran ( T ` b ) )
82		simprrr 805	. . . . . . . . . . . . . . . . . . . . . . 23 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( ( b e. W /\ c e. ran ( T ` b ) ) /\ ( o e. dom S /\ ( S ` o ) = b ) ) ) -> ( S ` o ) = b )
83	82	fveq2d 6195	. . . . . . . . . . . . . . . . . . . . . 22 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( ( b e. W /\ c e. ran ( T ` b ) ) /\ ( o e. dom S /\ ( S ` o ) = b ) ) ) -> ( T ` ( S ` o ) ) = ( T ` b ) )
84	83	rneqd 5353	. . . . . . . . . . . . . . . . . . . . 21 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( ( b e. W /\ c e. ran ( T ` b ) ) /\ ( o e. dom S /\ ( S ` o ) = b ) ) ) -> ran ( T ` ( S ` o ) ) = ran ( T ` b ) )
85	81, 84	eleqtrrd 2704	. . . . . . . . . . . . . . . . . . . 20 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( ( b e. W /\ c e. ran ( T ` b ) ) /\ ( o e. dom S /\ ( S ` o ) = b ) ) ) -> c e. ran ( T ` ( S ` o ) ) )
86	1, 2, 3, 4, 5, 6	efgsp1 18150	. . . . . . . . . . . . . . . . . . . 20 |- ( ( o e. dom S /\ c e. ran ( T ` ( S ` o ) ) ) -> ( o ++ <" c "> ) e. dom S )
87	75, 85, 86	syl2anc 693	. . . . . . . . . . . . . . . . . . 19 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( ( b e. W /\ c e. ran ( T ` b ) ) /\ ( o e. dom S /\ ( S ` o ) = b ) ) ) -> ( o ++ <" c "> ) e. dom S )
88	1, 2, 3, 4, 5, 6	efgsval2 18146	. . . . . . . . . . . . . . . . . . 19 |- ( ( o e. Word W /\ c e. W /\ ( o ++ <" c "> ) e. dom S ) -> ( S ` ( o ++ <" c "> ) ) = c )
89	79, 80, 87, 88	syl3anc 1326	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( ( b e. W /\ c e. ran ( T ` b ) ) /\ ( o e. dom S /\ ( S ` o ) = b ) ) ) -> ( S ` ( o ++ <" c "> ) ) = c )
90		fnfvelrn 6356	. . . . . . . . . . . . . . . . . . 19 |- ( ( S Fn dom S /\ ( o ++ <" c "> ) e. dom S ) -> ( S ` ( o ++ <" c "> ) ) e. ran S )
91	71, 87, 90	sylancr 695	. . . . . . . . . . . . . . . . . 18 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( ( b e. W /\ c e. ran ( T ` b ) ) /\ ( o e. dom S /\ ( S ` o ) = b ) ) ) -> ( S ` ( o ++ <" c "> ) ) e. ran S )
92	89, 91	eqeltrrd 2702	. . . . . . . . . . . . . . . . 17 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( ( b e. W /\ c e. ran ( T ` b ) ) /\ ( o e. dom S /\ ( S ` o ) = b ) ) ) -> c e. ran S )
93	92	anassrs 680	. . . . . . . . . . . . . . . 16 |- ( ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( b e. W /\ c e. ran ( T ` b ) ) ) /\ ( o e. dom S /\ ( S ` o ) = b ) ) -> c e. ran S )
94	74, 93	rexlimddv 3035	. . . . . . . . . . . . . . 15 |- ( ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) /\ ( b e. W /\ c e. ran ( T ` b ) ) ) -> c e. ran S )
95	94	rexlimdvaa 3032	. . . . . . . . . . . . . 14 |- ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) -> ( E. b e. W c e. ran ( T ` b ) -> c e. ran S ) )
96	50, 95	syl5bi 232	. . . . . . . . . . . . 13 |- ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) -> ( c e. U_ x e. W ran ( T ` x ) -> c e. ran S ) )
97		eldif 3584	. . . . . . . . . . . . . . . . . . 19 |- ( c e. ( W \ U_ x e. W ran ( T ` x ) ) <-> ( c e. W /\ -. c e. U_ x e. W ran ( T ` x ) ) )
98	5	eleq2i 2693	. . . . . . . . . . . . . . . . . . . 20 |- ( c e. D <-> c e. ( W \ U_ x e. W ran ( T ` x ) ) )
99	1, 2, 3, 4, 5, 6	efgs1 18148	. . . . . . . . . . . . . . . . . . . 20 |- ( c e. D -> <" c "> e. dom S )
100	98, 99	sylbir 225	. . . . . . . . . . . . . . . . . . 19 |- ( c e. ( W \ U_ x e. W ran ( T ` x ) ) -> <" c "> e. dom S )
101	97, 100	sylbir 225	. . . . . . . . . . . . . . . . . 18 |- ( ( c e. W /\ -. c e. U_ x e. W ran ( T ` x ) ) -> <" c "> e. dom S )
102	1, 2, 3, 4, 5, 6	efgsval 18144	. . . . . . . . . . . . . . . . . 18 |- ( <" c "> e. dom S -> ( S ` <" c "> ) = ( <" c "> ` ( ( # ` <" c "> ) - 1 ) ) )
103	101, 102	syl 17	. . . . . . . . . . . . . . . . 17 |- ( ( c e. W /\ -. c e. U_ x e. W ran ( T ` x ) ) -> ( S ` <" c "> ) = ( <" c "> ` ( ( # ` <" c "> ) - 1 ) ) )
104		s1len 13385	. . . . . . . . . . . . . . . . . . . . 21 |- ( # ` <" c "> ) = 1
105	104	oveq1i 6660	. . . . . . . . . . . . . . . . . . . 20 |- ( ( # ` <" c "> ) - 1 ) = ( 1 - 1 )
106		1m1e0 11089	. . . . . . . . . . . . . . . . . . . 20 |- ( 1 - 1 ) = 0
107	105, 106	eqtri 2644	. . . . . . . . . . . . . . . . . . 19 |- ( ( # ` <" c "> ) - 1 ) = 0
108	107	fveq2i 6194	. . . . . . . . . . . . . . . . . 18 |- ( <" c "> ` ( ( # ` <" c "> ) - 1 ) ) = ( <" c "> ` 0 )
109	108	a1i 11	. . . . . . . . . . . . . . . . 17 |- ( ( c e. W /\ -. c e. U_ x e. W ran ( T ` x ) ) -> ( <" c "> ` ( ( # ` <" c "> ) - 1 ) ) = ( <" c "> ` 0 ) )
110		s1fv 13390	. . . . . . . . . . . . . . . . . 18 |- ( c e. W -> ( <" c "> ` 0 ) = c )
111	110	adantr 481	. . . . . . . . . . . . . . . . 17 |- ( ( c e. W /\ -. c e. U_ x e. W ran ( T ` x ) ) -> ( <" c "> ` 0 ) = c )
112	103, 109, 111	3eqtrd 2660	. . . . . . . . . . . . . . . 16 |- ( ( c e. W /\ -. c e. U_ x e. W ran ( T ` x ) ) -> ( S ` <" c "> ) = c )
113		fnfvelrn 6356	. . . . . . . . . . . . . . . . 17 |- ( ( S Fn dom S /\ <" c "> e. dom S ) -> ( S ` <" c "> ) e. ran S )
114	71, 101, 113	sylancr 695	. . . . . . . . . . . . . . . 16 |- ( ( c e. W /\ -. c e. U_ x e. W ran ( T ` x ) ) -> ( S ` <" c "> ) e. ran S )
115	112, 114	eqeltrrd 2702	. . . . . . . . . . . . . . 15 |- ( ( c e. W /\ -. c e. U_ x e. W ran ( T ` x ) ) -> c e. ran S )
116	115	ex 450	. . . . . . . . . . . . . 14 |- ( c e. W -> ( -. c e. U_ x e. W ran ( T ` x ) -> c e. ran S ) )
117	116	3ad2ant2 1083	. . . . . . . . . . . . 13 |- ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) -> ( -. c e. U_ x e. W ran ( T ` x ) -> c e. ran S ) )
118	96, 117	pm2.61d 170	. . . . . . . . . . . 12 |- ( ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) /\ c e. W /\ ( # ` c ) = d ) -> c e. ran S )
119	118	rabssdv 3682	. . . . . . . . . . 11 |- ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) -> { c e. W | ( # ` c ) = d } C_ ran S )
120	44, 119	syl5eqss 3649	. . . . . . . . . 10 |- ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) -> { a e. W | ( # ` a ) = d } C_ ran S )
121	41, 120	unssd 3789	. . . . . . . . 9 |- ( ( d e. NN0 /\ { a e. W | ( # ` a ) < d } C_ ran S ) -> ( { a e. W | ( # ` a ) < d } u. { a e. W | ( # ` a ) = d } ) C_ ran S )
122	121	ex 450	. . . . . . . 8 |- ( d e. NN0 -> ( { a e. W | ( # ` a ) < d } C_ ran S -> ( { a e. W | ( # ` a ) < d } u. { a e. W | ( # ` a ) = d } ) C_ ran S ) )
123		id 22	. . . . . . . . . . . . 13 |- ( d e. NN0 -> d e. NN0 )
124		nn0leltp1 11436	. . . . . . . . . . . . 13 |- ( ( ( # ` a ) e. NN0 /\ d e. NN0 ) -> ( ( # ` a ) <_ d <-> ( # ` a ) < ( d + 1 ) ) )
125	21, 123, 124	syl2anr 495	. . . . . . . . . . . 12 |- ( ( d e. NN0 /\ a e. W ) -> ( ( # ` a ) <_ d <-> ( # ` a ) < ( d + 1 ) ) )
126	21	nn0red 11352	. . . . . . . . . . . . 13 |- ( a e. W -> ( # ` a ) e. RR )
127		nn0re 11301	. . . . . . . . . . . . 13 |- ( d e. NN0 -> d e. RR )
128		leloe 10124	. . . . . . . . . . . . 13 |- ( ( ( # ` a ) e. RR /\ d e. RR ) -> ( ( # ` a ) <_ d <-> ( ( # ` a ) < d \/ ( # ` a ) = d ) ) )
129	126, 127, 128	syl2anr 495	. . . . . . . . . . . 12 |- ( ( d e. NN0 /\ a e. W ) -> ( ( # ` a ) <_ d <-> ( ( # ` a ) < d \/ ( # ` a ) = d ) ) )
130	125, 129	bitr3d 270	. . . . . . . . . . 11 |- ( ( d e. NN0 /\ a e. W ) -> ( ( # ` a ) < ( d + 1 ) <-> ( ( # ` a ) < d \/ ( # ` a ) = d ) ) )
131	130	rabbidva 3188	. . . . . . . . . 10 |- ( d e. NN0 -> { a e. W | ( # ` a ) < ( d + 1 ) } = { a e. W | ( ( # ` a ) < d \/ ( # ` a ) = d ) } )
132		unrab 3898	. . . . . . . . . 10 |- ( { a e. W | ( # ` a ) < d } u. { a e. W | ( # ` a ) = d } ) = { a e. W | ( ( # ` a ) < d \/ ( # ` a ) = d ) }
133	131, 132	syl6eqr 2674	. . . . . . . . 9 |- ( d e. NN0 -> { a e. W | ( # ` a ) < ( d + 1 ) } = ( { a e. W | ( # ` a ) < d } u. { a e. W | ( # ` a ) = d } ) )
134	133	sseq1d 3632	. . . . . . . 8 |- ( d e. NN0 -> ( { a e. W | ( # ` a ) < ( d + 1 ) } C_ ran S <-> ( { a e. W | ( # ` a ) < d } u. { a e. W | ( # ` a ) = d } ) C_ ran S ) )
135	122, 134	sylibrd 249	. . . . . . 7 |- ( d e. NN0 -> ( { a e. W | ( # ` a ) < d } C_ ran S -> { a e. W | ( # ` a ) < ( d + 1 ) } C_ ran S ) )
136	30, 33, 36, 39, 40, 135	nn0ind 11472	. . . . . 6 |- ( ( ( # ` c ) + 1 ) e. NN0 -> { a e. W | ( # ` a ) < ( ( # ` c ) + 1 ) } C_ ran S )
137	17, 18, 136	3syl 18	. . . . 5 |- ( c e. W -> { a e. W | ( # ` a ) < ( ( # ` c ) + 1 ) } C_ ran S )
138		id 22	. . . . . 6 |- ( c e. W -> c e. W )
139	17	nn0red 11352	. . . . . . 7 |- ( c e. W -> ( # ` c ) e. RR )
140	139	ltp1d 10954	. . . . . 6 |- ( c e. W -> ( # ` c ) < ( ( # ` c ) + 1 ) )
141	42	breq1d 4663	. . . . . . 7 |- ( a = c -> ( ( # ` a ) < ( ( # ` c ) + 1 ) <-> ( # ` c ) < ( ( # ` c ) + 1 ) ) )
142	141	elrab 3363	. . . . . 6 |- ( c e. { a e. W | ( # ` a ) < ( ( # ` c ) + 1 ) } <-> ( c e. W /\ ( # ` c ) < ( ( # ` c ) + 1 ) ) )
143	138, 140, 142	sylanbrc 698	. . . . 5 |- ( c e. W -> c e. { a e. W | ( # ` a ) < ( ( # ` c ) + 1 ) } )
144	137, 143	sseldd 3604	. . . 4 |- ( c e. W -> c e. ran S )
145	144	ssriv 3607	. . 3 |- W C_ ran S
146	12, 145	eqssi 3619	. 2 |- ran S = W
147		dffo2 6119	. 2 |- ( S : dom S -onto-> W <-> ( S : dom S --> W /\ ran S = W ) )
148	10, 146, 147	mpbir2an 955	1 |- S : dom S -onto-> W

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   {crab 2916   \ cdif 3571   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   <.cop 4183   <.cotp 4185   U_ciun 4520   class class class wbr 4653   |-> cmpt 4729   _I cid 5023   X. cxp 5112   dom cdm 5114   ran crn 5115   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   1oc1o 7553   2oc2o 7554   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   2c2 11070   NN0cn0 11292   RR+crp 11832   ...cfz 12326  ..^cfzo 12465   #chash 13117  Word cword 13291   ++ cconcat 13293   <"cs1 13294   splice csplice 13296   <"cs2 13586   ~_FG cefg 18119

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-ot 4186  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-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  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-rp 11833  df-fz 12327  df-fzo 12466  df-hash 13118  df-word 13299  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-s2 13593

This theorem is referenced by:  efgredlemc  18158  efgrelexlemb  18163  efgredeu  18165  efgred2  18166