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Theorem fnsnsplit 5922
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 5546 . . 3  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
21adantr 452 . 2  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( F  |`  A )  =  F )
3 resundi 5152 . . 3  |-  ( F  |`  ( ( A  \  { X } )  u. 
{ X } ) )  =  ( ( F  |`  ( A  \  { X } ) )  u.  ( F  |`  { X } ) )
4 difsnid 3936 . . . . 5  |-  ( X  e.  A  ->  (
( A  \  { X } )  u.  { X } )  =  A )
54adantl 453 . . . 4  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( ( A  \  { X } )  u. 
{ X } )  =  A )
65reseq2d 5138 . . 3  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( F  |`  (
( A  \  { X } )  u.  { X } ) )  =  ( F  |`  A ) )
7 fnressn 5910 . . . 4  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( F  |`  { X } )  =  { <. X ,  ( F `
 X ) >. } )
87uneq2d 3493 . . 3  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( ( F  |`  ( A  \  { X } ) )  u.  ( F  |`  { X } ) )  =  ( ( F  |`  ( A  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
93, 6, 83eqtr3a 2491 . 2  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( F  |`  A )  =  ( ( F  |`  ( A  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
102, 9eqtr3d 2469 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 1652    e. wcel 1725    \ cdif 3309    u. cun 3310   {csn 3806   <.cop 3809    |` cres 4872    Fn wfn 5441   ` cfv 5446
This theorem is referenced by:  ralxpmap  26696
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454
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