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Theorem fargshiftfo 24765
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 5801 . . 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 24763 . . 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 5258 . . 3  |-  ran  G  =  { y  |  E. x  e.  ( 0..^ ( # `  F
) ) y  =  ( F `  (
x  +  1 ) ) }
6 fofn 5803 . . . . . 6  |-  ( F : ( 1 ... N ) -onto-> dom  E  ->  F  Fn  ( 1 ... N ) )
7 fnrnfv 5919 . . . . . 6  |-  ( F  Fn  ( 1 ... N )  ->  ran  F  =  { y  |  E. z  e.  ( 1 ... N ) y  =  ( F `
 z ) } )
86, 7syl 16 . . . . 5  |-  ( F : ( 1 ... N ) -onto-> dom  E  ->  ran  F  =  {
y  |  E. z  e.  ( 1 ... N
) y  =  ( F `  z ) } )
98adantl 466 . . . 4  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  ran  F  =  { y  |  E. z  e.  ( 1 ... N ) y  =  ( F `  z ) } )
10 df-fo 5600 . . . . . . 7  |-  ( F : ( 1 ... N ) -onto-> dom  E  <->  ( F  Fn  ( 1 ... N )  /\  ran  F  =  dom  E
) )
1110biimpi 194 . . . . . 6  |-  ( F : ( 1 ... N ) -onto-> dom  E  ->  ( F  Fn  (
1 ... N )  /\  ran  F  =  dom  E
) )
1211adantl 466 . . . . 5  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  ( F  Fn  ( 1 ... N
)  /\  ran  F  =  dom  E ) )
13 eqeq1 2461 . . . . . . . . 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 ) } ) )
14 eqcom 2466 . . . . . . . . 9  |-  ( dom 
E  =  { y  |  E. z  e.  ( 1 ... N
) y  =  ( F `  z ) }  <->  { y  |  E. z  e.  ( 1 ... N ) y  =  ( F `  z ) }  =  dom  E )
1513, 14syl6bb 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 ) )
16 ffn 5737 . . . . . . . . . . . . . 14  |-  ( F : ( 1 ... N ) --> dom  E  ->  F  Fn  ( 1 ... N ) )
17 fseq1hash 12447 . . . . . . . . . . . . . 14  |-  ( ( N  e.  NN0  /\  F  Fn  ( 1 ... N ) )  ->  ( # `  F
)  =  N )
1816, 17sylan2 474 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom  E
)  ->  ( # `  F
)  =  N )
191, 18sylan2 474 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  ( # `  F
)  =  N )
20 fargshiftlem 24761 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  x  e.  ( 0..^ N ) )  -> 
( x  +  1 )  e.  ( 1 ... N ) )
21 nn0z 10908 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  N  e.  ZZ )
22 fzval3 11888 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  ZZ  ->  (
1 ... N )  =  ( 1..^ ( N  +  1 ) ) )
2321, 22syl 16 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  ( 1 ... N )  =  ( 1..^ ( N  +  1 ) ) )
24 nn0cn 10826 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  N  e.  CC )
25 1cnd 9629 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN0  ->  1  e.  CC )
2624, 25addcomd 9799 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  NN0  ->  ( N  +  1 )  =  ( 1  +  N
) )
2726oveq2d 6312 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN0  ->  ( 1..^ ( N  +  1 ) )  =  ( 1..^ ( 1  +  N ) ) )
2823, 27eqtrd 2498 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  NN0  ->  ( 1 ... N )  =  ( 1..^ ( 1  +  N ) ) )
2928eleq2d 2527 . . . . . . . . . . . . . . . . . 18  |-  ( N  e.  NN0  ->  ( z  e.  ( 1 ... N )  <->  z  e.  ( 1..^ ( 1  +  N ) ) ) )
3029biimpa 484 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  z  e.  ( 1..^ ( 1  +  N ) ) )
3121adantr 465 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  N  e.  ZZ )
32 fzosubel3 11880 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  ( 1..^ ( 1  +  N
) )  /\  N  e.  ZZ )  ->  (
z  -  1 )  e.  ( 0..^ N ) )
3330, 31, 32syl2anc 661 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  ( z  - 
1 )  e.  ( 0..^ N ) )
34 oveq1 6303 . . . . . . . . . . . . . . . . . 18  |-  ( x  =  ( z  - 
1 )  ->  (
x  +  1 )  =  ( ( z  -  1 )  +  1 ) )
3534eqeq2d 2471 . . . . . . . . . . . . . . . . 17  |-  ( x  =  ( z  - 
1 )  ->  (
z  =  ( x  +  1 )  <->  z  =  ( ( z  - 
1 )  +  1 ) ) )
3635adantl 466 . . . . . . . . . . . . . . . 16  |-  ( ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  /\  x  =  ( z  -  1 ) )  ->  ( z  =  ( x  + 
1 )  <->  z  =  ( ( z  - 
1 )  +  1 ) ) )
37 elfzelz 11713 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  e.  ( 1 ... N )  ->  z  e.  ZZ )
3837zcnd 10991 . . . . . . . . . . . . . . . . . . 19  |-  ( z  e.  ( 1 ... N )  ->  z  e.  CC )
3938adantl 466 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  z  e.  CC )
40 1cnd 9629 . . . . . . . . . . . . . . . . . 18  |-  ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  1  e.  CC )
4139, 40npcand 9954 . . . . . . . . . . . . . . . . 17  |-  ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  ( ( z  -  1 )  +  1 )  =  z )
4241eqcomd 2465 . . . . . . . . . . . . . . . 16  |-  ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  z  =  ( ( z  -  1 )  +  1 ) )
4333, 36, 42rspcedvd 3215 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  z  e.  ( 1 ... N ) )  ->  E. x  e.  ( 0..^ N ) z  =  ( x  + 
1 ) )
44 fveq2 5872 . . . . . . . . . . . . . . . . 17  |-  ( z  =  ( x  + 
1 )  ->  ( F `  z )  =  ( F `  ( x  +  1
) ) )
4544eqeq2d 2471 . . . . . . . . . . . . . . . 16  |-  ( z  =  ( x  + 
1 )  ->  (
y  =  ( F `
 z )  <->  y  =  ( F `  ( x  +  1 ) ) ) )
4645adantl 466 . . . . . . . . . . . . . . 15  |-  ( ( N  e.  NN0  /\  z  =  ( x  +  1 ) )  ->  ( y  =  ( F `  z
)  <->  y  =  ( F `  ( x  +  1 ) ) ) )
4720, 43, 46rexxfrd 4671 . . . . . . . . . . . . . 14  |-  ( N  e.  NN0  ->  ( E. z  e.  ( 1 ... N ) y  =  ( F `  z )  <->  E. x  e.  ( 0..^ N ) y  =  ( F `
 ( x  + 
1 ) ) ) )
4847adantr 465 . . . . . . . . . . . . 13  |-  ( ( N  e.  NN0  /\  ( # `  F )  =  N )  -> 
( E. z  e.  ( 1 ... N
) y  =  ( F `  z )  <->  E. x  e.  (
0..^ N ) y  =  ( F `  ( x  +  1
) ) ) )
49 oveq2 6304 . . . . . . . . . . . . . . . 16  |-  ( (
# `  F )  =  N  ->  ( 0..^ ( # `  F
) )  =  ( 0..^ N ) )
5049rexeqdv 3061 . . . . . . . . . . . . . . 15  |-  ( (
# `  F )  =  N  ->  ( E. x  e.  ( 0..^ ( # `  F
) ) y  =  ( F `  (
x  +  1 ) )  <->  E. x  e.  ( 0..^ N ) y  =  ( F `  ( x  +  1
) ) ) )
5150bibi2d 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 ) ) ) ) )
5251adantl 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 ) ) ) ) )
5348, 52mpbird 232 . . . . . . . . . . . 12  |-  ( ( N  e.  NN0  /\  ( # `  F )  =  N )  -> 
( E. z  e.  ( 1 ... N
) y  =  ( F `  z )  <->  E. x  e.  (
0..^ ( # `  F
) ) y  =  ( F `  (
x  +  1 ) ) ) )
5419, 53syldan 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
) ) ) )
5554abbidv 2593 . . . . . . . . . 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 ) ) } )
5655eqeq1d 2459 . . . . . . . . 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 ) )
5756biimpcd 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 ) )
5815, 57syl6bi 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 ) ) )
5958com23 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 ) ) )
6059adantl 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 ) ) )
6112, 60mpcom 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 ) )
629, 61mpd 15 . . 3  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  { y  |  E. x  e.  ( 0..^ ( # `  F
) ) y  =  ( F `  (
x  +  1 ) ) }  =  dom  E )
635, 62syl5eq 2510 . 2  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) -onto-> dom  E
)  ->  ran  G  =  dom  E )
64 dffo2 5805 . 2  |-  ( G : ( 0..^ (
# `  F )
) -onto-> dom  E  <->  ( G : ( 0..^ (
# `  F )
) --> dom  E  /\  ran  G  =  dom  E
) )
654, 63, 64sylanbrc 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 1395    e. wcel 1819   {cab 2442   E.wrex 2808    |-> cmpt 4515   dom cdm 5008   ran crn 5009    Fn wfn 5589   -->wf 5590   -onto->wfo 5592   ` cfv 5594  (class class class)co 6296   CCcc 9507   0cc0 9509   1c1 9510    + caddc 9512    - cmin 9824   NN0cn0 10816   ZZcz 10885   ...cfz 11697  ..^cfzo 11821   #chash 12408
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-1o 7148  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-fin 7539  df-card 8337  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-fzo 11822  df-hash 12409
This theorem is referenced by:  eupatrl  25095
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