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Theorem fnsnsplit 5717
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 5353 . . 3  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
21adantr 451 . 2  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( F  |`  A )  =  F )
3 resundi 4969 . . 3  |-  ( F  |`  ( ( A  \  { X } )  u. 
{ X } ) )  =  ( ( F  |`  ( A  \  { X } ) )  u.  ( F  |`  { X } ) )
4 difsnid 3761 . . . . 5  |-  ( X  e.  A  ->  (
( A  \  { X } )  u.  { X } )  =  A )
54adantl 452 . . . 4  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( ( A  \  { X } )  u. 
{ X } )  =  A )
65reseq2d 4955 . . 3  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( F  |`  (
( A  \  { X } )  u.  { X } ) )  =  ( F  |`  A ) )
7 fnressn 5705 . . . 4  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( F  |`  { X } )  =  { <. X ,  ( F `
 X ) >. } )
87uneq2d 3329 . . 3  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( ( F  |`  ( A  \  { X } ) )  u.  ( F  |`  { X } ) )  =  ( ( F  |`  ( A  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
93, 6, 83eqtr3a 2339 . 2  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( F  |`  A )  =  ( ( F  |`  ( A  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
102, 9eqtr3d 2317 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 358    = wceq 1623    e. wcel 1684    \ cdif 3149    u. cun 3150   {csn 3640   <.cop 3643    |` cres 4691    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  ralxpmap  26761
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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