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Theorem fargshiftf 24298
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 5722 . . . 4  |-  ( F : ( 1 ... N ) --> dom  E  ->  F  Fn  ( 1 ... N ) )
2 fseq1hash 12399 . . . 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 2532 . . . . . 6  |-  ( N  =  ( # `  F
)  ->  ( N  e.  NN0  <->  ( # `  F
)  e.  NN0 )
)
5 oveq2 6283 . . . . . . 7  |-  ( N  =  ( # `  F
)  ->  ( 1 ... N )  =  ( 1 ... ( # `
 F ) ) )
65feq2d 5709 . . . . . 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 2472 . . . 4  |-  ( (
# `  F )  =  N  ->  ( ( N  e.  NN0  /\  F : ( 1 ... N ) --> dom  E
)  <->  ( ( # `  F )  e.  NN0  /\  F : ( 1 ... ( # `  F
) ) --> dom  E
) ) )
9 fargshiftlem 24296 . . . . . . 7  |-  ( ( ( # `  F
)  e.  NN0  /\  x  e.  ( 0..^ ( # `  F
) ) )  -> 
( x  +  1 )  e.  ( 1 ... ( # `  F
) ) )
10 ffvelrn 6010 . . . . . . . 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 6033 . 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 1374    e. wcel 1762   A.wral 2807    |-> cmpt 4498   dom cdm 4992    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275   0cc0 9481   1c1 9482    + caddc 9484   NN0cn0 10784   ...cfz 11661  ..^cfzo 11781   #chash 12360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6672  df-1st 6774  df-2nd 6775  df-recs 7032  df-rdg 7066  df-1o 7120  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-hash 12361
This theorem is referenced by:  fargshiftf1  24299  fargshiftfo  24300
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