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Theorem ccatfn 12556
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 12510 . 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 2467 . . . 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 3667 . . . . . . 7  |-  ( ran  s  u.  { (/) } )  C_  ( ( ran  s  u.  { (/) } )  u.  ( ran  t  u.  { (/) } ) )
4 fvrn0 5888 . . . . . . 7  |-  ( s `
 x )  e.  ( ran  s  u. 
{ (/) } )
53, 4sselii 3501 . . . . . 6  |-  ( s `
 x )  e.  ( ( ran  s  u.  { (/) } )  u.  ( ran  t  u. 
{ (/) } ) )
6 ssun2 3668 . . . . . . 7  |-  ( ran  t  u.  { (/) } )  C_  ( ( ran  s  u.  { (/) } )  u.  ( ran  t  u.  { (/) } ) )
7 fvrn0 5888 . . . . . . 7  |-  ( t `
 ( x  -  ( # `  s ) ) )  e.  ( ran  t  u.  { (/)
} )
86, 7sselii 3501 . . . . . 6  |-  ( t `
 ( x  -  ( # `  s ) ) )  e.  ( ( ran  s  u. 
{ (/) } )  u.  ( ran  t  u. 
{ (/) } ) )
95, 8keepel 4007 . . . . 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 6044 . . 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 6309 . . 3  |-  ( 0..^ ( ( # `  s
)  +  ( # `  t ) ) )  e.  _V
13 vex 3116 . . . . . 6  |-  s  e. 
_V
1413rnex 6718 . . . . 5  |-  ran  s  e.  _V
15 p0ex 4634 . . . . 5  |-  { (/) }  e.  _V
1614, 15unex 6582 . . . 4  |-  ( ran  s  u.  { (/) } )  e.  _V
17 vex 3116 . . . . . 6  |-  t  e. 
_V
1817rnex 6718 . . . . 5  |-  ran  t  e.  _V
1918, 15unex 6582 . . . 4  |-  ( ran  t  u.  { (/) } )  e.  _V
2016, 19unex 6582 . . 3  |-  ( ( ran  s  u.  { (/)
} )  u.  ( ran  t  u.  { (/) } ) )  e.  _V
21 fex2 6739 . . 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 1324 . 2  |-  ( x  e.  ( 0..^ ( ( # `  s
)  +  ( # `  t ) ) ) 
|->  if ( x  e.  ( 0..^ ( # `  s ) ) ,  ( s `  x
) ,  ( t `
 ( x  -  ( # `  s ) ) ) ) )  e.  _V
231, 22fnmpt2i 6853 1  |- concat  Fn  ( _V  X.  _V )
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1767   _Vcvv 3113    u. cun 3474   (/)c0 3785   ifcif 3939   {csn 4027    |-> cmpt 4505    X. cxp 4997   ran crn 5000    Fn wfn 5583   -->wf 5584   ` cfv 5588  (class class class)co 6284   0cc0 9492    + caddc 9495    - cmin 9805  ..^cfzo 11792   #chash 12373   concat cconcat 12502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-concat 12510
This theorem is referenced by:  frmdplusg  15854
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