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Theorem fargshiftf 24763
Description: If a class is a function, then also its "shifted function" is a function. (Contributed by Alexander van der Vekens, 23-Nov-2017.)
Hypothesis
Ref Expression
fargshift.g  |-  G  =  ( x  e.  ( 0..^ ( # `  F
) )  |->  ( F `
 ( x  + 
1 ) ) )
Assertion
Ref Expression
fargshiftf  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom  E
)  ->  G :
( 0..^ ( # `  F ) ) --> dom 
E )
Distinct variable groups:    x, F    x, E
Allowed substitution hints:    G( x)    N( x)

Proof of Theorem fargshiftf
StepHypRef Expression
1 ffn 5737 . . . 4  |-  ( F : ( 1 ... N ) --> dom  E  ->  F  Fn  ( 1 ... N ) )
2 fseq1hash 12447 . . . 4  |-  ( ( N  e.  NN0  /\  F  Fn  ( 1 ... N ) )  ->  ( # `  F
)  =  N )
31, 2sylan2 474 . . 3  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom  E
)  ->  ( # `  F
)  =  N )
4 eleq1 2529 . . . . . 6  |-  ( N  =  ( # `  F
)  ->  ( N  e.  NN0  <->  ( # `  F
)  e.  NN0 )
)
5 oveq2 6304 . . . . . . 7  |-  ( N  =  ( # `  F
)  ->  ( 1 ... N )  =  ( 1 ... ( # `
 F ) ) )
65feq2d 5724 . . . . . 6  |-  ( N  =  ( # `  F
)  ->  ( F : ( 1 ... N ) --> dom  E  <->  F : ( 1 ... ( # `  F
) ) --> dom  E
) )
74, 6anbi12d 710 . . . . 5  |-  ( N  =  ( # `  F
)  ->  ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom  E
)  <->  ( ( # `  F )  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) --> dom  E
) ) )
87eqcoms 2469 . . . 4  |-  ( (
# `  F )  =  N  ->  ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom  E
)  <->  ( ( # `  F )  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) --> dom  E
) ) )
9 fargshiftlem 24761 . . . . . . 7  |-  ( ( ( # `  F
)  e.  NN0  /\  x  e.  ( 0..^ ( # `  F
) ) )  -> 
( x  +  1 )  e.  ( 1 ... ( # `  F
) ) )
10 ffvelrn 6030 . . . . . . . 8  |-  ( ( F : ( 1 ... ( # `  F
) ) --> dom  E  /\  ( x  +  1 )  e.  ( 1 ... ( # `  F
) ) )  -> 
( F `  (
x  +  1 ) )  e.  dom  E
)
1110expcom 435 . . . . . . 7  |-  ( ( x  +  1 )  e.  ( 1 ... ( # `  F
) )  ->  ( F : ( 1 ... ( # `  F
) ) --> dom  E  ->  ( F `  (
x  +  1 ) )  e.  dom  E
) )
129, 11syl 16 . . . . . 6  |-  ( ( ( # `  F
)  e.  NN0  /\  x  e.  ( 0..^ ( # `  F
) ) )  -> 
( F : ( 1 ... ( # `  F ) ) --> dom 
E  ->  ( F `  ( x  +  1 ) )  e.  dom  E ) )
1312impancom 440 . . . . 5  |-  ( ( ( # `  F
)  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) --> dom  E
)  ->  ( x  e.  ( 0..^ ( # `  F ) )  -> 
( F `  (
x  +  1 ) )  e.  dom  E
) )
1413ralrimiv 2869 . . . 4  |-  ( ( ( # `  F
)  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) --> dom  E
)  ->  A. x  e.  ( 0..^ ( # `  F ) ) ( F `  ( x  +  1 ) )  e.  dom  E )
158, 14syl6bi 228 . . 3  |-  ( (
# `  F )  =  N  ->  ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom  E
)  ->  A. x  e.  ( 0..^ ( # `  F ) ) ( F `  ( x  +  1 ) )  e.  dom  E ) )
163, 15mpcom 36 . 2  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom  E
)  ->  A. x  e.  ( 0..^ ( # `  F ) ) ( F `  ( x  +  1 ) )  e.  dom  E )
17 fargshift.g . . 3  |-  G  =  ( x  e.  ( 0..^ ( # `  F
) )  |->  ( F `
 ( x  + 
1 ) ) )
1817fmpt 6053 . 2  |-  ( A. x  e.  ( 0..^ ( # `  F
) ) ( F `
 ( x  + 
1 ) )  e. 
dom  E  <->  G : ( 0..^ ( # `  F
) ) --> dom  E
)
1916, 18sylib 196 1  |-  ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom  E
)  ->  G :
( 0..^ ( # `  F ) ) --> dom 
E )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   A.wral 2807    |-> cmpt 4515   dom cdm 5008    Fn wfn 5589   -->wf 5590   ` cfv 5594  (class class class)co 6296   0cc0 9509   1c1 9510    + caddc 9512   NN0cn0 10816   ...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:  fargshiftf1  24764  fargshiftfo  24765
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