Theorem shftfn 11847

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 4964	. . . . 5 |- Rel { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) }
2	1	a1i 11	. . . 4 |- ( ( F Fn B /\ A e. CC ) -> Rel { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
3		fnfun 5505	. . . . . 6 |- ( F Fn B -> Fun F )
4	3	adantr 452	. . . . 5 |- ( ( F Fn B /\ A e. CC ) -> Fun F )
5		funmo 5433	. . . . . . 7 |- ( Fun F -> E* w ( z - A ) F w )
6		vex 2923	. . . . . . . . . 10 |- z e. _V
7		vex 2923	. . . . . . . . . 10 |- w e. _V
8		eleq1 2468	. . . . . . . . . . 11 |- ( x = z -> ( x e. CC <-> z e. CC ) )
9		oveq1 6051	. . . . . . . . . . . 12 |- ( x = z -> ( x - A ) = ( z - A ) )
10	9	breq1d 4186	. . . . . . . . . . 11 |- ( x = z -> ( ( x - A ) F y <-> ( z - A ) F y ) )
11	8, 10	anbi12d 692	. . . . . . . . . 10 |- ( x = z -> ( ( x e. CC /\ ( x - A ) F y ) <-> ( z e. CC /\ ( z - A ) F y ) ) )
12		breq2 4180	. . . . . . . . . . 11 |- ( y = w -> ( ( z - A ) F y <-> ( z - A ) F w ) )
13	12	anbi2d 685	. . . . . . . . . 10 |- ( y = w -> ( ( z e. CC /\ ( z - A ) F y ) <-> ( z e. CC /\ ( z - A ) F w ) ) )
14		eqid 2408	. . . . . . . . . 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 4441	. . . . . . . . 9 |- ( z { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } w <-> ( z e. CC /\ ( z - A ) F w ) )
16	15	simprbi 451	. . . . . . . 8 |- ( z { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } w -> ( z - A ) F w )
17	16	moimi 2305	. . . . . . 7 |- ( E* w ( z - A ) F w -> E* w z { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } w )
18	5, 17	syl 16	. . . . . 6 |- ( Fun F -> E* w z { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } w )
19	18	alrimiv 1638	. . . . 5 |- ( Fun F -> A. z E* w z { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } w )
20	4, 19	syl 16	. . . 4 |- ( ( F Fn B /\ A e. CC ) -> A. z E* w z { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } w )
21		dffun6 5432	. . . 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 646	. . 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 11844	. . . . 5 |- ( A e. CC -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
25	24	adantl 453	. . . 4 |- ( ( F Fn B /\ A e. CC ) -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
26	25	funeqd 5438	. . 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 224	. 2 |- ( ( F Fn B /\ A e. CC ) -> Fun ( F shift A ) )
28	23	shftdm 11845	. . 3 |- ( A e. CC -> dom ( F shift A ) = { x e. CC | ( x - A ) e. dom F } )
29		fndm 5507	. . . . 5 |- ( F Fn B -> dom F = B )
30	29	eleq2d 2475	. . . 4 |- ( F Fn B -> ( ( x - A ) e. dom F <-> ( x - A ) e. B ) )
31	30	rabbidv 2912	. . 3 |- ( F Fn B -> { x e. CC | ( x - A ) e. dom F } = { x e. CC | ( x - A ) e. B } )
32	28, 31	sylan9eqr 2462	. 2 |- ( ( F Fn B /\ A e. CC ) -> dom ( F shift A ) = { x e. CC | ( x - A ) e. B } )
33		df-fn 5420	. 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 646	1 |- ( ( F Fn B /\ A e. CC ) -> ( F shift A ) Fn { x e. CC | ( x - A ) e. B } )

Syntax hints:   -> wi 4   /\ wa 359   A.wal 1546   = wceq 1649   e. wcel 1721   E*wmo 2259   {crab 2674   _Vcvv 2920   class class class wbr 4176   {copab 4229   dom cdm 4841   Rel wrel 4846   Fun wfun 5411   Fn wfn 5412  (class class class)co 6044   CCcc 8948   - cmin 9251   shift cshi 11840

This theorem is referenced by:  shftf  11853  seqshft  11859

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664  ax-resscn 9007  ax-1cn 9008  ax-icn 9009  ax-addcl 9010  ax-addrcl 9011  ax-mulcl 9012  ax-mulrcl 9013  ax-mulcom 9014  ax-addass 9015  ax-mulass 9016  ax-distr 9017  ax-i2m1 9018  ax-1ne0 9019  ax-1rid 9020  ax-rnegex 9021  ax-rrecex 9022  ax-cnre 9023  ax-pre-lttri 9024  ax-pre-lttrn 9025  ax-pre-ltadd 9026

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-nel 2574  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-po 4467  df-so 4468  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6047  df-oprab 6048  df-mpt2 6049  df-riota 6512  df-er 6868  df-en 7073  df-dom 7074  df-sdom 7075  df-pnf 9082  df-mnf 9083  df-ltxr 9085  df-sub 9253  df-shft 11841