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Theorem ccatfn 11696
Description: The concatenation operator is a two-argument function. (Contributed by Mario Carneiro, 27-Sep-2015.)
Assertion
Ref Expression
ccatfn  |- concat  Fn  ( _V  X.  _V )

Proof of Theorem ccatfn
Dummy variables  t 
s  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-concat 11679 . 2  |- concat  =  ( s  e.  _V , 
t  e.  _V  |->  ( x  e.  ( 0..^ ( ( # `  s
)  +  ( # `  t ) ) ) 
|->  if ( x  e.  ( 0..^ ( # `  s ) ) ,  ( s `  x
) ,  ( t `
 ( x  -  ( # `  s ) ) ) ) ) )
2 eqid 2404 . . . 4  |-  ( x  e.  ( 0..^ ( ( # `  s
)  +  ( # `  t ) ) ) 
|->  if ( x  e.  ( 0..^ ( # `  s ) ) ,  ( s `  x
) ,  ( t `
 ( x  -  ( # `  s ) ) ) ) )  =  ( x  e.  ( 0..^ ( (
# `  s )  +  ( # `  t
) ) )  |->  if ( x  e.  ( 0..^ ( # `  s
) ) ,  ( s `  x ) ,  ( t `  ( x  -  ( # `
 s ) ) ) ) )
3 ssun1 3470 . . . . . . 7  |-  ( ran  s  u.  { (/) } )  C_  ( ( ran  s  u.  { (/) } )  u.  ( ran  t  u.  { (/) } ) )
4 fvrn0 5712 . . . . . . 7  |-  ( s `
 x )  e.  ( ran  s  u. 
{ (/) } )
53, 4sselii 3305 . . . . . 6  |-  ( s `
 x )  e.  ( ( ran  s  u.  { (/) } )  u.  ( ran  t  u. 
{ (/) } ) )
6 ssun2 3471 . . . . . . 7  |-  ( ran  t  u.  { (/) } )  C_  ( ( ran  s  u.  { (/) } )  u.  ( ran  t  u.  { (/) } ) )
7 fvrn0 5712 . . . . . . 7  |-  ( t `
 ( x  -  ( # `  s ) ) )  e.  ( ran  t  u.  { (/)
} )
86, 7sselii 3305 . . . . . 6  |-  ( t `
 ( x  -  ( # `  s ) ) )  e.  ( ( ran  s  u. 
{ (/) } )  u.  ( ran  t  u. 
{ (/) } ) )
95, 8keepel 3756 . . . . 5  |-  if ( x  e.  ( 0..^ ( # `  s
) ) ,  ( s `  x ) ,  ( t `  ( x  -  ( # `
 s ) ) ) )  e.  ( ( ran  s  u. 
{ (/) } )  u.  ( ran  t  u. 
{ (/) } ) )
109a1i 11 . . . 4  |-  ( x  e.  ( 0..^ ( ( # `  s
)  +  ( # `  t ) ) )  ->  if ( x  e.  ( 0..^ (
# `  s )
) ,  ( s `
 x ) ,  ( t `  (
x  -  ( # `  s ) ) ) )  e.  ( ( ran  s  u.  { (/)
} )  u.  ( ran  t  u.  { (/) } ) ) )
112, 10fmpti 5851 . . 3  |-  ( x  e.  ( 0..^ ( ( # `  s
)  +  ( # `  t ) ) ) 
|->  if ( x  e.  ( 0..^ ( # `  s ) ) ,  ( s `  x
) ,  ( t `
 ( x  -  ( # `  s ) ) ) ) ) : ( 0..^ ( ( # `  s
)  +  ( # `  t ) ) ) --> ( ( ran  s  u.  { (/) } )  u.  ( ran  t  u. 
{ (/) } ) )
12 ovex 6065 . . 3  |-  ( 0..^ ( ( # `  s
)  +  ( # `  t ) ) )  e.  _V
13 vex 2919 . . . . . 6  |-  s  e. 
_V
1413rnex 5092 . . . . 5  |-  ran  s  e.  _V
15 p0ex 4346 . . . . 5  |-  { (/) }  e.  _V
1614, 15unex 4666 . . . 4  |-  ( ran  s  u.  { (/) } )  e.  _V
17 vex 2919 . . . . . 6  |-  t  e. 
_V
1817rnex 5092 . . . . 5  |-  ran  t  e.  _V
1918, 15unex 4666 . . . 4  |-  ( ran  t  u.  { (/) } )  e.  _V
2016, 19unex 4666 . . 3  |-  ( ( ran  s  u.  { (/)
} )  u.  ( ran  t  u.  { (/) } ) )  e.  _V
21 fex2 5562 . . 3  |-  ( ( ( x  e.  ( 0..^ ( ( # `  s )  +  (
# `  t )
) )  |->  if ( x  e.  ( 0..^ ( # `  s
) ) ,  ( s `  x ) ,  ( t `  ( x  -  ( # `
 s ) ) ) ) ) : ( 0..^ ( (
# `  s )  +  ( # `  t
) ) ) --> ( ( ran  s  u. 
{ (/) } )  u.  ( ran  t  u. 
{ (/) } ) )  /\  ( 0..^ ( ( # `  s
)  +  ( # `  t ) ) )  e.  _V  /\  (
( ran  s  u.  {
(/) } )  u.  ( ran  t  u.  { (/) } ) )  e.  _V )  ->  ( x  e.  ( 0..^ ( (
# `  s )  +  ( # `  t
) ) )  |->  if ( x  e.  ( 0..^ ( # `  s
) ) ,  ( s `  x ) ,  ( t `  ( x  -  ( # `
 s ) ) ) ) )  e. 
_V )
2211, 12, 20, 21mp3an 1279 . 2  |-  ( x  e.  ( 0..^ ( ( # `  s
)  +  ( # `  t ) ) ) 
|->  if ( x  e.  ( 0..^ ( # `  s ) ) ,  ( s `  x
) ,  ( t `
 ( x  -  ( # `  s ) ) ) ) )  e.  _V
231, 22fnmpt2i 6379 1  |- concat  Fn  ( _V  X.  _V )
Colors of variables: wff set class
Syntax hints:    e. wcel 1721   _Vcvv 2916    u. cun 3278   (/)c0 3588   ifcif 3699   {csn 3774    e. cmpt 4226    X. cxp 4835   ran crn 4838    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040   0cc0 8946    + caddc 8949    - cmin 9247  ..^cfzo 11090   #chash 11573   concat cconcat 11673
This theorem is referenced by:  frmdplusg  14754
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-concat 11679
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