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Theorem ccatfn 12291
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 12250 . 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 2443 . . . 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 3538 . . . . . . 7  |-  ( ran  s  u.  { (/) } )  C_  ( ( ran  s  u.  { (/) } )  u.  ( ran  t  u.  { (/) } ) )
4 fvrn0 5731 . . . . . . 7  |-  ( s `
 x )  e.  ( ran  s  u. 
{ (/) } )
53, 4sselii 3372 . . . . . 6  |-  ( s `
 x )  e.  ( ( ran  s  u.  { (/) } )  u.  ( ran  t  u. 
{ (/) } ) )
6 ssun2 3539 . . . . . . 7  |-  ( ran  t  u.  { (/) } )  C_  ( ( ran  s  u.  { (/) } )  u.  ( ran  t  u.  { (/) } ) )
7 fvrn0 5731 . . . . . . 7  |-  ( t `
 ( x  -  ( # `  s ) ) )  e.  ( ran  t  u.  { (/)
} )
86, 7sselii 3372 . . . . . 6  |-  ( t `
 ( x  -  ( # `  s ) ) )  e.  ( ( ran  s  u. 
{ (/) } )  u.  ( ran  t  u. 
{ (/) } ) )
95, 8keepel 3876 . . . . 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 5885 . . 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 6135 . . 3  |-  ( 0..^ ( ( # `  s
)  +  ( # `  t ) ) )  e.  _V
13 vex 2994 . . . . . 6  |-  s  e. 
_V
1413rnex 6531 . . . . 5  |-  ran  s  e.  _V
15 p0ex 4498 . . . . 5  |-  { (/) }  e.  _V
1614, 15unex 6397 . . . 4  |-  ( ran  s  u.  { (/) } )  e.  _V
17 vex 2994 . . . . . 6  |-  t  e. 
_V
1817rnex 6531 . . . . 5  |-  ran  t  e.  _V
1918, 15unex 6397 . . . 4  |-  ( ran  t  u.  { (/) } )  e.  _V
2016, 19unex 6397 . . 3  |-  ( ( ran  s  u.  { (/)
} )  u.  ( ran  t  u.  { (/) } ) )  e.  _V
21 fex2 6551 . . 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 1314 . 2  |-  ( x  e.  ( 0..^ ( ( # `  s
)  +  ( # `  t ) ) ) 
|->  if ( x  e.  ( 0..^ ( # `  s ) ) ,  ( s `  x
) ,  ( t `
 ( x  -  ( # `  s ) ) ) ) )  e.  _V
231, 22fnmpt2i 6662 1  |- concat  Fn  ( _V  X.  _V )
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1756   _Vcvv 2991    u. cun 3345   (/)c0 3656   ifcif 3810   {csn 3896    e. cmpt 4369    X. cxp 4857   ran crn 4860    Fn wfn 5432   -->wf 5433   ` cfv 5437  (class class class)co 6110   0cc0 9301    + caddc 9304    - cmin 9614  ..^cfzo 11567   #chash 12122   concat cconcat 12242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550  ax-un 6391
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-pw 3881  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-fv 5445  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6596  df-2nd 6597  df-concat 12250
This theorem is referenced by:  frmdplusg  15551
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