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Theorem fnsnsplitdc 6454
Description: Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.) (Revised by Jim Kingdon, 29-Jan-2023.)
Assertion
Ref Expression
fnsnsplitdc  |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  F  Fn  A  /\  X  e.  A
)  ->  F  =  ( ( F  |`  ( A  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
Distinct variable groups:    x, A, y   
x, X, y
Allowed substitution hints:    F( x, y)

Proof of Theorem fnsnsplitdc
StepHypRef Expression
1 fnresdm 5281 . . 3  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
213ad2ant2 1004 . 2  |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  F  Fn  A  /\  X  e.  A
)  ->  ( F  |`  A )  =  F )
3 resundi 4881 . . 3  |-  ( F  |`  ( ( A  \  { X } )  u. 
{ X } ) )  =  ( ( F  |`  ( A  \  { X } ) )  u.  ( F  |`  { X } ) )
4 dcdifsnid 6453 . . . . 5  |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  X  e.  A )  ->  (
( A  \  { X } )  u.  { X } )  =  A )
543adant2 1001 . . . 4  |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  F  Fn  A  /\  X  e.  A
)  ->  ( ( A  \  { X }
)  u.  { X } )  =  A )
65reseq2d 4868 . . 3  |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  F  Fn  A  /\  X  e.  A
)  ->  ( F  |`  ( ( A  \  { X } )  u. 
{ X } ) )  =  ( F  |`  A ) )
7 fnressn 5655 . . . . 5  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( F  |`  { X } )  =  { <. X ,  ( F `
 X ) >. } )
87uneq2d 3262 . . . 4  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( ( F  |`  ( A  \  { X } ) )  u.  ( F  |`  { X } ) )  =  ( ( F  |`  ( A  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
983adant1 1000 . . 3  |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  F  Fn  A  /\  X  e.  A
)  ->  ( ( F  |`  ( A  \  { X } ) )  u.  ( F  |`  { X } ) )  =  ( ( F  |`  ( A  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
103, 6, 93eqtr3a 2214 . 2  |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  F  Fn  A  /\  X  e.  A
)  ->  ( F  |`  A )  =  ( ( F  |`  ( A  \  { X }
) )  u.  { <. X ,  ( F `
 X ) >. } ) )
112, 10eqtr3d 2192 1  |-  ( ( A. x  e.  A  A. y  e.  A DECID  x  =  y  /\  F  Fn  A  /\  X  e.  A
)  ->  F  =  ( ( F  |`  ( A  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103  DECID wdc 820    /\ w3a 963    = wceq 1335    e. wcel 2128   A.wral 2435    \ cdif 3099    u. cun 3100   {csn 3561   <.cop 3564    |` cres 4590    Fn wfn 5167   ` cfv 5172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4084  ax-pow 4137  ax-pr 4171
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-reu 2442  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-br 3968  df-opab 4028  df-id 4255  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-iota 5137  df-fun 5174  df-fn 5175  df-f 5176  df-f1 5177  df-fo 5178  df-f1o 5179  df-fv 5180
This theorem is referenced by:  funresdfunsndc  6455
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