Theorem shftfn 10766

Description: Functionality and domain of a sequence shifted by A. (Contributed by NM, 20-Jul-2005.) (Revised by Mario Carneiro, 3-Nov-2013.)

Hypothesis

Ref	Expression
shftfval.1	|- F e. _V

Assertion

Ref	Expression
shftfn	|- ( ( F Fn B /\ A e. CC ) -> ( F shift A ) Fn { x e. CC | ( x - A ) e. B } )

Distinct variable groups:   x, A   x, F   x, B

Proof of Theorem shftfn

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relopab 4731	. . . . 5 |- Rel { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) }
2	1	a1i 9	. . . 4 |- ( ( F Fn B /\ A e. CC ) -> Rel { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
3		fnfun 5285	. . . . . 6 |- ( F Fn B -> Fun F )
4	3	adantr 274	. . . . 5 |- ( ( F Fn B /\ A e. CC ) -> Fun F )
5		funmo 5203	. . . . . . 7 |- ( Fun F -> E* w ( z - A ) F w )
6		vex 2729	. . . . . . . . . 10 |- z e. _V
7		vex 2729	. . . . . . . . . 10 |- w e. _V
8		eleq1 2229	. . . . . . . . . . 11 |- ( x = z -> ( x e. CC <-> z e. CC ) )
9		oveq1 5849	. . . . . . . . . . . 12 |- ( x = z -> ( x - A ) = ( z - A ) )
10	9	breq1d 3992	. . . . . . . . . . 11 |- ( x = z -> ( ( x - A ) F y <-> ( z - A ) F y ) )
11	8, 10	anbi12d 465	. . . . . . . . . 10 |- ( x = z -> ( ( x e. CC /\ ( x - A ) F y ) <-> ( z e. CC /\ ( z - A ) F y ) ) )
12		breq2 3986	. . . . . . . . . . 11 |- ( y = w -> ( ( z - A ) F y <-> ( z - A ) F w ) )
13	12	anbi2d 460	. . . . . . . . . 10 |- ( y = w -> ( ( z e. CC /\ ( z - A ) F y ) <-> ( z e. CC /\ ( z - A ) F w ) ) )
14		eqid 2165	. . . . . . . . . 10 |- { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) }
15	6, 7, 11, 13, 14	brab 4250	. . . . . . . . 9 |- ( z { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } w <-> ( z e. CC /\ ( z - A ) F w ) )
16	15	simprbi 273	. . . . . . . 8 |- ( z { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } w -> ( z - A ) F w )
17	16	moimi 2079	. . . . . . 7 |- ( E* w ( z - A ) F w -> E* w z { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } w )
18	5, 17	syl 14	. . . . . 6 |- ( Fun F -> E* w z { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } w )
19	18	alrimiv 1862	. . . . 5 |- ( Fun F -> A. z E* w z { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } w )
20	4, 19	syl 14	. . . 4 |- ( ( F Fn B /\ A e. CC ) -> A. z E* w z { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } w )
21		dffun6 5202	. . . 4 |- ( Fun { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } <-> ( Rel { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } /\ A. z E* w z { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } w ) )
22	2, 20, 21	sylanbrc 414	. . 3 |- ( ( F Fn B /\ A e. CC ) -> Fun { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
23		shftfval.1	. . . . . 6 |- F e. _V
24	23	shftfval 10763	. . . . 5 |- ( A e. CC -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
25	24	adantl 275	. . . 4 |- ( ( F Fn B /\ A e. CC ) -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
26	25	funeqd 5210	. . 3 |- ( ( F Fn B /\ A e. CC ) -> ( Fun ( F shift A ) <-> Fun { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } ) )
27	22, 26	mpbird 166	. 2 |- ( ( F Fn B /\ A e. CC ) -> Fun ( F shift A ) )
28	23	shftdm 10764	. . 3 |- ( A e. CC -> dom ( F shift A ) = { x e. CC | ( x - A ) e. dom F } )
29		fndm 5287	. . . . 5 |- ( F Fn B -> dom F = B )
30	29	eleq2d 2236	. . . 4 |- ( F Fn B -> ( ( x - A ) e. dom F <-> ( x - A ) e. B ) )
31	30	rabbidv 2715	. . 3 |- ( F Fn B -> { x e. CC | ( x - A ) e. dom F } = { x e. CC | ( x - A ) e. B } )
32	28, 31	sylan9eqr 2221	. 2 |- ( ( F Fn B /\ A e. CC ) -> dom ( F shift A ) = { x e. CC | ( x - A ) e. B } )
33		df-fn 5191	. 2 |- ( ( F shift A ) Fn { x e. CC | ( x - A ) e. B } <-> ( Fun ( F shift A ) /\ dom ( F shift A ) = { x e. CC | ( x - A ) e. B } ) )
34	27, 32, 33	sylanbrc 414	1 |- ( ( F Fn B /\ A e. CC ) -> ( F shift A ) Fn { x e. CC | ( x - A ) e. B } )

Syntax hints:   -> wi 4   /\ wa 103   A.wal 1341   = wceq 1343   E*wmo 2015   e. wcel 2136   {crab 2448   _Vcvv 2726   class class class wbr 3982   {copab 4042   dom cdm 4604   Rel wrel 4609   Fun wfun 5182   Fn wfn 5183  (class class class)co 5842   CCcc 7751   - cmin 8069   shift cshi 10756

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-sub 8071  df-shft 10757

This theorem is referenced by:  shftf  10772  seq3shft  10780