MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ccatfn Structured version   Unicode version

Theorem ccatfn 12268
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 12227 . 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 2441 . . . 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 3516 . . . . . . 7  |-  ( ran  s  u.  { (/) } )  C_  ( ( ran  s  u.  { (/) } )  u.  ( ran  t  u.  { (/) } ) )
4 fvrn0 5709 . . . . . . 7  |-  ( s `
 x )  e.  ( ran  s  u. 
{ (/) } )
53, 4sselii 3350 . . . . . 6  |-  ( s `
 x )  e.  ( ( ran  s  u.  { (/) } )  u.  ( ran  t  u. 
{ (/) } ) )
6 ssun2 3517 . . . . . . 7  |-  ( ran  t  u.  { (/) } )  C_  ( ( ran  s  u.  { (/) } )  u.  ( ran  t  u.  { (/) } ) )
7 fvrn0 5709 . . . . . . 7  |-  ( t `
 ( x  -  ( # `  s ) ) )  e.  ( ran  t  u.  { (/)
} )
86, 7sselii 3350 . . . . . 6  |-  ( t `
 ( x  -  ( # `  s ) ) )  e.  ( ( ran  s  u. 
{ (/) } )  u.  ( ran  t  u. 
{ (/) } ) )
95, 8keepel 3854 . . . . 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 5863 . . 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 6115 . . 3  |-  ( 0..^ ( ( # `  s
)  +  ( # `  t ) ) )  e.  _V
13 vex 2973 . . . . . 6  |-  s  e. 
_V
1413rnex 6511 . . . . 5  |-  ran  s  e.  _V
15 p0ex 4476 . . . . 5  |-  { (/) }  e.  _V
1614, 15unex 6377 . . . 4  |-  ( ran  s  u.  { (/) } )  e.  _V
17 vex 2973 . . . . . 6  |-  t  e. 
_V
1817rnex 6511 . . . . 5  |-  ran  t  e.  _V
1918, 15unex 6377 . . . 4  |-  ( ran  t  u.  { (/) } )  e.  _V
2016, 19unex 6377 . . 3  |-  ( ( ran  s  u.  { (/)
} )  u.  ( ran  t  u.  { (/) } ) )  e.  _V
21 fex2 6531 . . 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 1309 . 2  |-  ( x  e.  ( 0..^ ( ( # `  s
)  +  ( # `  t ) ) ) 
|->  if ( x  e.  ( 0..^ ( # `  s ) ) ,  ( s `  x
) ,  ( t `
 ( x  -  ( # `  s ) ) ) ) )  e.  _V
231, 22fnmpt2i 6642 1  |- concat  Fn  ( _V  X.  _V )
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1761   _Vcvv 2970    u. cun 3323   (/)c0 3634   ifcif 3788   {csn 3874    e. cmpt 4347    X. cxp 4834   ran crn 4837    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   0cc0 9278    + caddc 9281    - cmin 9591  ..^cfzo 11544   #chash 12099   concat cconcat 12219
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 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-concat 12227
This theorem is referenced by:  frmdplusg  15525
  Copyright terms: Public domain W3C validator