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Theorem fargshiftfo 24342
Description: If a function is onto, then also the shifted function is onto. (Contributed by Alexander van der Vekens, 24-Nov-2017.)
Hypothesis
Ref Expression
fargshift.g  |-  G  =  ( x  e.  ( 0..^ ( # `  F
) )  |->  ( F `
 ( x  + 
1 ) ) )
Assertion
Ref Expression
fargshiftfo  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  G :
( 0..^ ( # `  F ) ) -onto-> dom 
E )
Distinct variable groups:    x, F    x, E    x, N
Allowed substitution hint:    G( x)

Proof of Theorem fargshiftfo
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fof 5795 . . 3  |-  ( F : ( 1 ... N ) -onto-> dom  E  ->  F : ( 1 ... N ) --> dom 
E )
2 fargshift.g . . . 4  |-  G  =  ( x  e.  ( 0..^ ( # `  F
) )  |->  ( F `
 ( x  + 
1 ) ) )
32fargshiftf 24340 . . 3  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom  E
)  ->  G :
( 0..^ ( # `  F ) ) --> dom 
E )
41, 3sylan2 474 . 2  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  G :
( 0..^ ( # `  F ) ) --> dom 
E )
52rnmpt 5248 . . . 4  |-  ran  G  =  { y  |  E. x  e.  ( 0..^ ( # `  F
) ) y  =  ( F `  (
x  +  1 ) ) }
65a1i 11 . . 3  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  ran  G  =  { y  |  E. x  e.  ( 0..^ ( # `  F
) ) y  =  ( F `  (
x  +  1 ) ) } )
7 fofn 5797 . . . . . 6  |-  ( F : ( 1 ... N ) -onto-> dom  E  ->  F  Fn  ( 1 ... N ) )
8 fnrnfv 5914 . . . . . 6  |-  ( F  Fn  ( 1 ... N )  ->  ran  F  =  { y  |  E. z  e.  ( 1 ... N ) y  =  ( F `
 z ) } )
97, 8syl 16 . . . . 5  |-  ( F : ( 1 ... N ) -onto-> dom  E  ->  ran  F  =  {
y  |  E. z  e.  ( 1 ... N
) y  =  ( F `  z ) } )
109adantl 466 . . . 4  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  ran  F  =  { y  |  E. z  e.  ( 1 ... N ) y  =  ( F `  z ) } )
11 df-fo 5594 . . . . . . 7  |-  ( F : ( 1 ... N ) -onto-> dom  E  <->  ( F  Fn  ( 1 ... N )  /\  ran  F  =  dom  E
) )
1211biimpi 194 . . . . . 6  |-  ( F : ( 1 ... N ) -onto-> dom  E  ->  ( F  Fn  (
1 ... N )  /\  ran  F  =  dom  E
) )
1312adantl 466 . . . . 5  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  ( F  Fn  ( 1 ... N
)  /\  ran  F  =  dom  E ) )
14 eqeq1 2471 . . . . . . . . 9  |-  ( ran 
F  =  dom  E  ->  ( ran  F  =  { y  |  E. z  e.  ( 1 ... N ) y  =  ( F `  z ) }  <->  dom  E  =  { y  |  E. z  e.  ( 1 ... N ) y  =  ( F `  z ) } ) )
15 eqcom 2476 . . . . . . . . 9  |-  ( dom 
E  =  { y  |  E. z  e.  ( 1 ... N
) y  =  ( F `  z ) }  <->  { y  |  E. z  e.  ( 1 ... N ) y  =  ( F `  z ) }  =  dom  E )
1614, 15syl6bb 261 . . . . . . . 8  |-  ( ran 
F  =  dom  E  ->  ( ran  F  =  { y  |  E. z  e.  ( 1 ... N ) y  =  ( F `  z ) }  <->  { y  |  E. z  e.  ( 1 ... N ) y  =  ( F `
 z ) }  =  dom  E ) )
17 ffn 5731 . . . . . . . . . . . . . 14  |-  ( F : ( 1 ... N ) --> dom  E  ->  F  Fn  ( 1 ... N ) )
18 fseq1hash 12412 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  F  Fn  ( 1 ... N ) )  ->  ( # `  F
)  =  N )
1917, 18sylan2 474 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom  E
)  ->  ( # `  F
)  =  N )
201, 19sylan2 474 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  ( # `  F
)  =  N )
21 fargshiftlem 24338 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ( 0..^ N ) )  -> 
( x  +  1 )  e.  ( 1 ... N ) )
22 nn0z 10887 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN0  ->  N  e.  ZZ )
23 fzval3 11853 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  ZZ  ->  (
1 ... N )  =  ( 1..^ ( N  +  1 ) ) )
2422, 23syl 16 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  ( 1 ... N )  =  ( 1..^ ( N  +  1 ) ) )
25 nn0cn 10805 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  N  e.  CC )
26 ax-1cn 9550 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  1  e.  CC
2726a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  NN0  ->  1  e.  CC )
2825, 27addcomd 9781 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( N  e.  NN0  ->  ( N  +  1 )  =  ( 1  +  N
) )
2928oveq2d 6300 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  ( 1..^ ( N  +  1 ) )  =  ( 1..^ ( 1  +  N ) ) )
3024, 29eqtrd 2508 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  ( 1 ... N )  =  ( 1..^ ( 1  +  N ) ) )
3130eleq2d 2537 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  ( z  e.  ( 1 ... N )  <->  z  e.  ( 1..^ ( 1  +  N ) ) ) )
3231biimpa 484 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  z  e.  ( 1..^ ( 1  +  N ) ) )
3322adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  N  e.  ZZ )
34 fzosubel3 11845 . . . . . . . . . . . . . . . . . . 19  |-  ( ( z  e.  ( 1..^ ( 1  +  N
) )  /\  N  e.  ZZ )  ->  (
z  -  1 )  e.  ( 0..^ N ) )
3532, 33, 34syl2anc 661 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  ( z  - 
1 )  e.  ( 0..^ N ) )
36 elfzelz 11688 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( z  e.  ( 1 ... N )  ->  z  e.  ZZ )
3736zcnd 10967 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( z  e.  ( 1 ... N )  ->  z  e.  CC )
3837adantl 466 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  z  e.  CC )
3926a1i 11 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  1  e.  CC )
4038, 39npcand 9934 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  ( ( z  -  1 )  +  1 )  =  z )
4140eqcomd 2475 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  z  =  ( ( z  -  1 )  +  1 ) )
4241adantr 465 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  /\  x  =  ( z  -  1 ) )  ->  z  =  ( ( z  - 
1 )  +  1 ) )
43 oveq1 6291 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( x  =  ( z  - 
1 )  ->  (
x  +  1 )  =  ( ( z  -  1 )  +  1 ) )
4443eqeq2d 2481 . . . . . . . . . . . . . . . . . . . . 21  |-  ( x  =  ( z  - 
1 )  ->  (
z  =  ( x  +  1 )  <->  z  =  ( ( z  - 
1 )  +  1 ) ) )
4544adantl 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  /\  x  =  ( z  -  1 ) )  ->  ( z  =  ( x  + 
1 )  <->  z  =  ( ( z  - 
1 )  +  1 ) ) )
4642, 45mpbird 232 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  /\  x  =  ( z  -  1 ) )  ->  z  =  ( x  +  1
) )
4746a1d 25 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  /\  x  =  ( z  -  1 ) )  ->  ( z  e.  ( 1 ... N
)  ->  z  =  ( x  +  1
) ) )
4835, 47rspcimedv 3216 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  ( z  e.  ( 1 ... N
)  ->  E. x  e.  ( 0..^ N ) z  =  ( x  +  1 ) ) )
4948com12 31 . . . . . . . . . . . . . . . 16  |-  ( z  e.  ( 1 ... N )  ->  (
( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  E. x  e.  ( 0..^ N ) z  =  ( x  + 
1 ) ) )
5049anabsi7 817 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  E. x  e.  ( 0..^ N ) z  =  ( x  + 
1 ) )
51 fveq2 5866 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( x  + 
1 )  ->  ( F `  z )  =  ( F `  ( x  +  1
) ) )
5251eqeq2d 2481 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( x  + 
1 )  ->  (
y  =  ( F `
 z )  <->  y  =  ( F `  ( x  +  1 ) ) ) )
5352adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  z  =  ( x  +  1 ) )  ->  ( y  =  ( F `  z
)  <->  y  =  ( F `  ( x  +  1 ) ) ) )
5421, 50, 53rexxfrd 4662 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  ( E. z  e.  ( 1 ... N ) y  =  ( F `  z )  <->  E. x  e.  ( 0..^ N ) y  =  ( F `
 ( x  + 
1 ) ) ) )
5554adantr 465 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  ( # `  F )  =  N )  -> 
( E. z  e.  ( 1 ... N
) y  =  ( F `  z )  <->  E. x  e.  (
0..^ N ) y  =  ( F `  ( x  +  1
) ) ) )
56 oveq2 6292 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  N  ->  ( 0..^ ( # `  F
) )  =  ( 0..^ N ) )
5756rexeqdv 3065 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  N  ->  ( E. x  e.  ( 0..^ ( # `  F
) ) y  =  ( F `  (
x  +  1 ) )  <->  E. x  e.  ( 0..^ N ) y  =  ( F `  ( x  +  1
) ) ) )
5857bibi2d 318 . . . . . . . . . . . . . 14  |-  ( (
# `  F )  =  N  ->  ( ( E. z  e.  ( 1 ... N ) y  =  ( F `
 z )  <->  E. x  e.  ( 0..^ ( # `  F ) ) y  =  ( F `  ( x  +  1
) ) )  <->  ( E. z  e.  ( 1 ... N ) y  =  ( F `  z )  <->  E. x  e.  ( 0..^ N ) y  =  ( F `
 ( x  + 
1 ) ) ) ) )
5958adantl 466 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  ( # `  F )  =  N )  -> 
( ( E. z  e.  ( 1 ... N
) y  =  ( F `  z )  <->  E. x  e.  (
0..^ ( # `  F
) ) y  =  ( F `  (
x  +  1 ) ) )  <->  ( E. z  e.  ( 1 ... N ) y  =  ( F `  z )  <->  E. x  e.  ( 0..^ N ) y  =  ( F `
 ( x  + 
1 ) ) ) ) )
6055, 59mpbird 232 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  ( # `  F )  =  N )  -> 
( E. z  e.  ( 1 ... N
) y  =  ( F `  z )  <->  E. x  e.  (
0..^ ( # `  F
) ) y  =  ( F `  (
x  +  1 ) ) ) )
6120, 60syldan 470 . . . . . . . . . . 11  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  ( E. z  e.  ( 1 ... N ) y  =  ( F `  z )  <->  E. x  e.  ( 0..^ ( # `  F ) ) y  =  ( F `  ( x  +  1
) ) ) )
6261abbidv 2603 . . . . . . . . . 10  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  { y  |  E. z  e.  ( 1 ... N ) y  =  ( F `
 z ) }  =  { y  |  E. x  e.  ( 0..^ ( # `  F
) ) y  =  ( F `  (
x  +  1 ) ) } )
6362eqeq1d 2469 . . . . . . . . 9  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  ( {
y  |  E. z  e.  ( 1 ... N
) y  =  ( F `  z ) }  =  dom  E  <->  { y  |  E. x  e.  ( 0..^ ( # `  F ) ) y  =  ( F `  ( x  +  1
) ) }  =  dom  E ) )
6463biimpcd 224 . . . . . . . 8  |-  ( { y  |  E. z  e.  ( 1 ... N
) y  =  ( F `  z ) }  =  dom  E  ->  ( ( N  e. 
NN0  /\  F :
( 1 ... N
) -onto-> dom  E )  ->  { y  |  E. x  e.  ( 0..^ ( # `  F
) ) y  =  ( F `  (
x  +  1 ) ) }  =  dom  E ) )
6516, 64syl6bi 228 . . . . . . 7  |-  ( ran 
F  =  dom  E  ->  ( ran  F  =  { y  |  E. z  e.  ( 1 ... N ) y  =  ( F `  z ) }  ->  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom 
E )  ->  { y  |  E. x  e.  ( 0..^ ( # `  F ) ) y  =  ( F `  ( x  +  1
) ) }  =  dom  E ) ) )
6665com23 78 . . . . . 6  |-  ( ran 
F  =  dom  E  ->  ( ( N  e. 
NN0  /\  F :
( 1 ... N
) -onto-> dom  E )  -> 
( ran  F  =  { y  |  E. z  e.  ( 1 ... N ) y  =  ( F `  z ) }  ->  { y  |  E. x  e.  ( 0..^ ( # `  F ) ) y  =  ( F `  ( x  +  1
) ) }  =  dom  E ) ) )
6766adantl 466 . . . . 5  |-  ( ( F  Fn  ( 1 ... N )  /\  ran  F  =  dom  E
)  ->  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  ( ran  F  =  { y  |  E. z  e.  ( 1 ... N ) y  =  ( F `
 z ) }  ->  { y  |  E. x  e.  ( 0..^ ( # `  F
) ) y  =  ( F `  (
x  +  1 ) ) }  =  dom  E ) ) )
6813, 67mpcom 36 . . . 4  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  ( ran  F  =  { y  |  E. z  e.  ( 1 ... N ) y  =  ( F `
 z ) }  ->  { y  |  E. x  e.  ( 0..^ ( # `  F
) ) y  =  ( F `  (
x  +  1 ) ) }  =  dom  E ) )
6910, 68mpd 15 . . 3  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  { y  |  E. x  e.  ( 0..^ ( # `  F
) ) y  =  ( F `  (
x  +  1 ) ) }  =  dom  E )
706, 69eqtrd 2508 . 2  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  ran  G  =  dom  E )
71 dffo2 5799 . 2  |-  ( G : ( 0..^ (
# `  F )
) -onto-> dom  E  <->  ( G : ( 0..^ (
# `  F )
) --> dom  E  /\  ran  G  =  dom  E
) )
724, 70, 71sylanbrc 664 1  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  G :
( 0..^ ( # `  F ) ) -onto-> dom 
E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2815    |-> cmpt 4505   dom cdm 4999   ran crn 5000    Fn wfn 5583   -->wf 5584   -onto->wfo 5586   ` cfv 5588  (class class class)co 6284   CCcc 9490   0cc0 9492   1c1 9493    + caddc 9495    - cmin 9805   NN0cn0 10795   ZZcz 10864   ...cfz 11672  ..^cfzo 11792   #chash 12373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8320  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-fzo 11793  df-hash 12374
This theorem is referenced by:  eupatrl  24672
  Copyright terms: Public domain W3C validator