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Theorem fnsnsplit 5870
Description: Split a function into a single point and all the rest. (Contributed by Stefan O'Rear, 27-Feb-2015.)
Assertion
Ref Expression
fnsnsplit  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  F  =  ( ( F  |`  ( A  \  { X } ) )  u.  { <. X ,  ( F `  X ) >. } ) )

Proof of Theorem fnsnsplit
StepHypRef Expression
1 fnresdm 5495 . . 3  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
21adantr 452 . 2  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( F  |`  A )  =  F )
3 resundi 5101 . . 3  |-  ( F  |`  ( ( A  \  { X } )  u. 
{ X } ) )  =  ( ( F  |`  ( A  \  { X } ) )  u.  ( F  |`  { X } ) )
4 difsnid 3888 . . . . 5  |-  ( X  e.  A  ->  (
( A  \  { X } )  u.  { X } )  =  A )
54adantl 453 . . . 4  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( ( A  \  { X } )  u. 
{ X } )  =  A )
65reseq2d 5087 . . 3  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( F  |`  (
( A  \  { X } )  u.  { X } ) )  =  ( F  |`  A ) )
7 fnressn 5858 . . . 4  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( F  |`  { X } )  =  { <. X ,  ( F `
 X ) >. } )
87uneq2d 3445 . . 3  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( ( F  |`  ( A  \  { X } ) )  u.  ( F  |`  { X } ) )  =  ( ( F  |`  ( A  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
93, 6, 83eqtr3a 2444 . 2  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( F  |`  A )  =  ( ( F  |`  ( A  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
102, 9eqtr3d 2422 1  |-  ( ( 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 359    = wceq 1649    e. wcel 1717    \ cdif 3261    u. cun 3262   {csn 3758   <.cop 3761    |` cres 4821    Fn wfn 5390   ` cfv 5395
This theorem is referenced by:  ralxpmap  26434
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403
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