Theorem shftfn 9850

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 4492	. . . . 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 5027	. . . . . 6 |- ( F Fn B -> Fun F )
4	3	adantr 270	. . . . 5 |- ( ( F Fn B /\ A e. CC ) -> Fun F )
5		funmo 4947	. . . . . . 7 |- ( Fun F -> E* w ( z - A ) F w )
6		vex 2605	. . . . . . . . . 10 |- z e. _V
7		vex 2605	. . . . . . . . . 10 |- w e. _V
8		eleq1 2142	. . . . . . . . . . 11 |- ( x = z -> ( x e. CC <-> z e. CC ) )
9		oveq1 5550	. . . . . . . . . . . 12 |- ( x = z -> ( x - A ) = ( z - A ) )
10	9	breq1d 3803	. . . . . . . . . . 11 |- ( x = z -> ( ( x - A ) F y <-> ( z - A ) F y ) )
11	8, 10	anbi12d 457	. . . . . . . . . 10 |- ( x = z -> ( ( x e. CC /\ ( x - A ) F y ) <-> ( z e. CC /\ ( z - A ) F y ) ) )
12		breq2 3797	. . . . . . . . . . 11 |- ( y = w -> ( ( z - A ) F y <-> ( z - A ) F w ) )
13	12	anbi2d 452	. . . . . . . . . 10 |- ( y = w -> ( ( z e. CC /\ ( z - A ) F y ) <-> ( z e. CC /\ ( z - A ) F w ) ) )
14		eqid 2082	. . . . . . . . . 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 4035	. . . . . . . . 9 |- ( z { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } w <-> ( z e. CC /\ ( z - A ) F w ) )
16	15	simprbi 269	. . . . . . . 8 |- ( z { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } w -> ( z - A ) F w )
17	16	moimi 2007	. . . . . . 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 1796	. . . . 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 4946	. . . 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 408	. . 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 9847	. . . . 5 |- ( A e. CC -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
25	24	adantl 271	. . . 4 |- ( ( F Fn B /\ A e. CC ) -> ( F shift A ) = { <. x , y >. | ( x e. CC /\ ( x - A ) F y ) } )
26	25	funeqd 4953	. . 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 165	. 2 |- ( ( F Fn B /\ A e. CC ) -> Fun ( F shift A ) )
28	23	shftdm 9848	. . 3 |- ( A e. CC -> dom ( F shift A ) = { x e. CC | ( x - A ) e. dom F } )
29		fndm 5029	. . . . 5 |- ( F Fn B -> dom F = B )
30	29	eleq2d 2149	. . . 4 |- ( F Fn B -> ( ( x - A ) e. dom F <-> ( x - A ) e. B ) )
31	30	rabbidv 2594	. . 3 |- ( F Fn B -> { x e. CC | ( x - A ) e. dom F } = { x e. CC | ( x - A ) e. B } )
32	28, 31	sylan9eqr 2136	. 2 |- ( ( F Fn B /\ A e. CC ) -> dom ( F shift A ) = { x e. CC | ( x - A ) e. B } )
33		df-fn 4935	. 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 408	1 |- ( ( F Fn B /\ A e. CC ) -> ( F shift A ) Fn { x e. CC | ( x - A ) e. B } )

Syntax hints:   -> wi 4   /\ wa 102   A.wal 1283   = wceq 1285   e. wcel 1434   E*wmo 1943   {crab 2353   _Vcvv 2602   class class class wbr 3793   {copab 3846   dom cdm 4371   Rel wrel 4376   Fun wfun 4926   Fn wfn 4927  (class class class)co 5543   CCcc 7041   - cmin 7346   shift cshi 9840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-resscn 7130  ax-1cn 7131  ax-icn 7133  ax-addcl 7134  ax-addrcl 7135  ax-mulcl 7136  ax-addcom 7138  ax-addass 7140  ax-distr 7142  ax-i2m1 7143  ax-0id 7146  ax-rnegex 7147  ax-cnre 7149

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-riota 5499  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-sub 7348  df-shft 9841

This theorem is referenced by:  shftf  9856