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Theorem finxpnom 31863
Description: Cartesian exponentiation when the exponent is not a natural number defaults to the empty set. (Contributed by ML, 24-Oct-2020.)
Assertion
Ref Expression
finxpnom  |-  ( -.  N  e.  om  ->  ( U ^^ ^^ N
)  =  (/) )

Proof of Theorem finxpnom
Dummy variables  n  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 464 . . . . 5  |-  ( ( N  e.  om  /\  (/)  =  ( rec (
( n  e.  om ,  x  e.  _V  |->  if ( ( n  =  1o  /\  x  e.  U ) ,  (/) ,  if ( x  e.  ( _V  X.  U
) ,  <. U. n ,  ( 1st `  x
) >. ,  <. n ,  x >. ) ) ) ,  <. N ,  y
>. ) `  N ) )  ->  N  e.  om )
21con3i 142 . . . 4  |-  ( -.  N  e.  om  ->  -.  ( N  e.  om  /\  (/)  =  ( rec (
( n  e.  om ,  x  e.  _V  |->  if ( ( n  =  1o  /\  x  e.  U ) ,  (/) ,  if ( x  e.  ( _V  X.  U
) ,  <. U. n ,  ( 1st `  x
) >. ,  <. n ,  x >. ) ) ) ,  <. N ,  y
>. ) `  N ) ) )
3 abid 2459 . . . 4  |-  ( y  e.  { y  |  ( N  e.  om  /\  (/)  =  ( rec (
( n  e.  om ,  x  e.  _V  |->  if ( ( n  =  1o  /\  x  e.  U ) ,  (/) ,  if ( x  e.  ( _V  X.  U
) ,  <. U. n ,  ( 1st `  x
) >. ,  <. n ,  x >. ) ) ) ,  <. N ,  y
>. ) `  N ) ) }  <->  ( N  e.  om  /\  (/)  =  ( rec ( ( n  e.  om ,  x  e.  _V  |->  if ( ( n  =  1o  /\  x  e.  U ) ,  (/) ,  if ( x  e.  ( _V 
X.  U ) , 
<. U. n ,  ( 1st `  x )
>. ,  <. n ,  x >. ) ) ) ,  <. N ,  y
>. ) `  N ) ) )
42, 3sylnibr 312 . . 3  |-  ( -.  N  e.  om  ->  -.  y  e.  { y  |  ( N  e. 
om  /\  (/)  =  ( rec ( ( n  e.  om ,  x  e.  _V  |->  if ( ( n  =  1o  /\  x  e.  U ) ,  (/) ,  if ( x  e.  ( _V 
X.  U ) , 
<. U. n ,  ( 1st `  x )
>. ,  <. n ,  x >. ) ) ) ,  <. N ,  y
>. ) `  N ) ) } )
5 df-finxp 31846 . . . 4  |-  ( U ^^ ^^ N )  =  { y  |  ( N  e.  om  /\  (/)  =  ( rec (
( n  e.  om ,  x  e.  _V  |->  if ( ( n  =  1o  /\  x  e.  U ) ,  (/) ,  if ( x  e.  ( _V  X.  U
) ,  <. U. n ,  ( 1st `  x
) >. ,  <. n ,  x >. ) ) ) ,  <. N ,  y
>. ) `  N ) ) }
65eleq2i 2541 . . 3  |-  ( y  e.  ( U ^^ ^^ N )  <->  y  e.  { y  |  ( N  e.  om  /\  (/)  =  ( rec ( ( n  e.  om ,  x  e.  _V  |->  if ( ( n  =  1o  /\  x  e.  U ) ,  (/) ,  if ( x  e.  ( _V 
X.  U ) , 
<. U. n ,  ( 1st `  x )
>. ,  <. n ,  x >. ) ) ) ,  <. N ,  y
>. ) `  N ) ) } )
74, 6sylnibr 312 . 2  |-  ( -.  N  e.  om  ->  -.  y  e.  ( U ^^ ^^ N ) )
87eq0rdv 3773 1  |-  ( -.  N  e.  om  ->  ( U ^^ ^^ N
)  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   {cab 2457   _Vcvv 3031   (/)c0 3722   ifcif 3872   <.cop 3965   U.cuni 4190    X. cxp 4837   ` cfv 5589    |-> cmpt2 6310   omcom 6711   1stc1st 6810   reccrdg 7145   1oc1o 7193   ^^
^^cfinxp 31845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-dif 3393  df-in 3397  df-ss 3404  df-nul 3723  df-finxp 31846
This theorem is referenced by:  finxp00  31864
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