Theorem shftfn 10596

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 4666	. . . . 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 5220	. . . . . 6 |- ( F Fn B -> Fun F )
4	3	adantr 274	. . . . 5 |- ( ( F Fn B /\ A e. CC ) -> Fun F )
5		funmo 5138	. . . . . . 7 |- ( Fun F -> E* w ( z - A ) F w )
6		vex 2689	. . . . . . . . . 10 |- z e. _V
7		vex 2689	. . . . . . . . . 10 |- w e. _V
8		eleq1 2202	. . . . . . . . . . 11 |- ( x = z -> ( x e. CC <-> z e. CC ) )
9		oveq1 5781	. . . . . . . . . . . 12 |- ( x = z -> ( x - A ) = ( z - A ) )
10	9	breq1d 3939	. . . . . . . . . . 11 |- ( x = z -> ( ( x - A ) F y <-> ( z - A ) F y ) )
11	8, 10	anbi12d 464	. . . . . . . . . 10 |- ( x = z -> ( ( x e. CC /\ ( x - A ) F y ) <-> ( z e. CC /\ ( z - A ) F y ) ) )
12		breq2 3933	. . . . . . . . . . 11 |- ( y = w -> ( ( z - A ) F y <-> ( z - A ) F w ) )
13	12	anbi2d 459	. . . . . . . . . 10 |- ( y = w -> ( ( z e. CC /\ ( z - A ) F y ) <-> ( z e. CC /\ ( z - A ) F w ) ) )
14		eqid 2139	. . . . . . . . . 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 4194	. . . . . . . . 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 2064	. . . . . . 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 1846	. . . . 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 5137	. . . 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 413	. . 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 10593	. . . . 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 5145	. . 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 10594	. . 3 |- ( A e. CC -> dom ( F shift A ) = { x e. CC | ( x - A ) e. dom F } )
29		fndm 5222	. . . . 5 |- ( F Fn B -> dom F = B )
30	29	eleq2d 2209	. . . 4 |- ( F Fn B -> ( ( x - A ) e. dom F <-> ( x - A ) e. B ) )
31	30	rabbidv 2675	. . 3 |- ( F Fn B -> { x e. CC | ( x - A ) e. dom F } = { x e. CC | ( x - A ) e. B } )
32	28, 31	sylan9eqr 2194	. 2 |- ( ( F Fn B /\ A e. CC ) -> dom ( F shift A ) = { x e. CC | ( x - A ) e. B } )
33		df-fn 5126	. 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 413	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 1329   = wceq 1331   e. wcel 1480   E*wmo 2000   {crab 2420   _Vcvv 2686   class class class wbr 3929   {copab 3988   dom cdm 4539   Rel wrel 4544   Fun wfun 5117   Fn wfn 5118  (class class class)co 5774   CCcc 7618   - cmin 7933   shift cshi 10586

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-resscn 7712  ax-1cn 7713  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-addass 7722  ax-distr 7724  ax-i2m1 7725  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sub 7935  df-shft 10587

This theorem is referenced by:  shftf  10602  seq3shft  10610