Theorem shftfn 10971

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 4789	. . . . 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 5352	. . . . . 6 |- ( F Fn B -> Fun F )
4	3	adantr 276	. . . . 5 |- ( ( F Fn B /\ A e. CC ) -> Fun F )
5		funmo 5270	. . . . . . 7 |- ( Fun F -> E* w ( z - A ) F w )
6		vex 2763	. . . . . . . . . 10 |- z e. _V
7		vex 2763	. . . . . . . . . 10 |- w e. _V
8		eleq1 2256	. . . . . . . . . . 11 |- ( x = z -> ( x e. CC <-> z e. CC ) )
9		oveq1 5926	. . . . . . . . . . . 12 |- ( x = z -> ( x - A ) = ( z - A ) )
10	9	breq1d 4040	. . . . . . . . . . 11 |- ( x = z -> ( ( x - A ) F y <-> ( z - A ) F y ) )
11	8, 10	anbi12d 473	. . . . . . . . . 10 |- ( x = z -> ( ( x e. CC /\ ( x - A ) F y ) <-> ( z e. CC /\ ( z - A ) F y ) ) )
12		breq2 4034	. . . . . . . . . . 11 |- ( y = w -> ( ( z - A ) F y <-> ( z - A ) F w ) )
13	12	anbi2d 464	. . . . . . . . . 10 |- ( y = w -> ( ( z e. CC /\ ( z - A ) F y ) <-> ( z e. CC /\ ( z - A ) F w ) ) )
14		eqid 2193	. . . . . . . . . 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 4304	. . . . . . . . 9 |- ( z { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } w <-> ( z e. CC /\ ( z - A ) F w ) )
16	15	simprbi 275	. . . . . . . 8 |- ( z { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } w -> ( z - A ) F w )
17	16	moimi 2107	. . . . . . 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 1885	. . . . 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 5269	. . . 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 417	. . 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 10968	. . . . 5 |- ( A e. CC -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
25	24	adantl 277	. . . 4 |- ( ( F Fn B /\ A e. CC ) -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
26	25	funeqd 5277	. . 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 167	. 2 |- ( ( F Fn B /\ A e. CC ) -> Fun ( F shift A ) )
28	23	shftdm 10969	. . 3 |- ( A e. CC -> dom ( F shift A ) = { x e. CC | ( x - A ) e. dom F } )
29		fndm 5354	. . . . 5 |- ( F Fn B -> dom F = B )
30	29	eleq2d 2263	. . . 4 |- ( F Fn B -> ( ( x - A ) e. dom F <-> ( x - A ) e. B ) )
31	30	rabbidv 2749	. . 3 |- ( F Fn B -> { x e. CC | ( x - A ) e. dom F } = { x e. CC | ( x - A ) e. B } )
32	28, 31	sylan9eqr 2248	. 2 |- ( ( F Fn B /\ A e. CC ) -> dom ( F shift A ) = { x e. CC | ( x - A ) e. B } )
33		df-fn 5258	. 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 417	1 |- ( ( F Fn B /\ A e. CC ) -> ( F shift A ) Fn { x e. CC | ( x - A ) e. B } )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1362   = wceq 1364   E*wmo 2043   e. wcel 2164   {crab 2476   _Vcvv 2760   class class class wbr 4030   {copab 4090   dom cdm 4660   Rel wrel 4665   Fun wfun 5249   Fn wfn 5250  (class class class)co 5919   CCcc 7872   - cmin 8192   shift cshi 10961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-resscn 7966  ax-1cn 7967  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-addcom 7974  ax-addass 7976  ax-distr 7978  ax-i2m1 7979  ax-0id 7982  ax-rnegex 7983  ax-cnre 7985

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-sub 8194  df-shft 10962

This theorem is referenced by:  shftf  10977  seq3shft  10985