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Theorem ccatfn 11741
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 11724 . 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 2436 . . . 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 3510 . . . . . . 7  |-  ( ran  s  u.  { (/) } )  C_  ( ( ran  s  u.  { (/) } )  u.  ( ran  t  u.  { (/) } ) )
4 fvrn0 5753 . . . . . . 7  |-  ( s `
 x )  e.  ( ran  s  u. 
{ (/) } )
53, 4sselii 3345 . . . . . 6  |-  ( s `
 x )  e.  ( ( ran  s  u.  { (/) } )  u.  ( ran  t  u. 
{ (/) } ) )
6 ssun2 3511 . . . . . . 7  |-  ( ran  t  u.  { (/) } )  C_  ( ( ran  s  u.  { (/) } )  u.  ( ran  t  u.  { (/) } ) )
7 fvrn0 5753 . . . . . . 7  |-  ( t `
 ( x  -  ( # `  s ) ) )  e.  ( ran  t  u.  { (/)
} )
86, 7sselii 3345 . . . . . 6  |-  ( t `
 ( x  -  ( # `  s ) ) )  e.  ( ( ran  s  u. 
{ (/) } )  u.  ( ran  t  u. 
{ (/) } ) )
95, 8keepel 3796 . . . . 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 5892 . . 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 6106 . . 3  |-  ( 0..^ ( ( # `  s
)  +  ( # `  t ) ) )  e.  _V
13 vex 2959 . . . . . 6  |-  s  e. 
_V
1413rnex 5133 . . . . 5  |-  ran  s  e.  _V
15 p0ex 4386 . . . . 5  |-  { (/) }  e.  _V
1614, 15unex 4707 . . . 4  |-  ( ran  s  u.  { (/) } )  e.  _V
17 vex 2959 . . . . . 6  |-  t  e. 
_V
1817rnex 5133 . . . . 5  |-  ran  t  e.  _V
1918, 15unex 4707 . . . 4  |-  ( ran  t  u.  { (/) } )  e.  _V
2016, 19unex 4707 . . 3  |-  ( ( ran  s  u.  { (/)
} )  u.  ( ran  t  u.  { (/) } ) )  e.  _V
21 fex2 5603 . . 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 6420 1  |- concat  Fn  ( _V  X.  _V )
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   _Vcvv 2956    u. cun 3318   (/)c0 3628   ifcif 3739   {csn 3814    e. cmpt 4266    X. cxp 4876   ran crn 4879    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081   0cc0 8990    + caddc 8993    - cmin 9291  ..^cfzo 11135   #chash 11618   concat cconcat 11718
This theorem is referenced by:  frmdplusg  14799
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-concat 11724
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