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Theorem fnsnsplit 6014
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 5618 . . 3  |-  ( F  Fn  A  ->  ( F  |`  A )  =  F )
21adantr 465 . 2  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( F  |`  A )  =  F )
3 resundi 5222 . . 3  |-  ( F  |`  ( ( A  \  { X } )  u. 
{ X } ) )  =  ( ( F  |`  ( A  \  { X } ) )  u.  ( F  |`  { X } ) )
4 difsnid 4117 . . . . 5  |-  ( X  e.  A  ->  (
( A  \  { X } )  u.  { X } )  =  A )
54adantl 466 . . . 4  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( ( A  \  { X } )  u. 
{ X } )  =  A )
65reseq2d 5208 . . 3  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( F  |`  (
( A  \  { X } )  u.  { X } ) )  =  ( F  |`  A ) )
7 fnressn 5993 . . . 4  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( F  |`  { X } )  =  { <. X ,  ( F `
 X ) >. } )
87uneq2d 3608 . . 3  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( ( F  |`  ( A  \  { X } ) )  u.  ( F  |`  { X } ) )  =  ( ( F  |`  ( A  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
93, 6, 83eqtr3a 2516 . 2  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  ( F  |`  A )  =  ( ( F  |`  ( A  \  { X } ) )  u. 
{ <. X ,  ( F `  X )
>. } ) )
102, 9eqtr3d 2494 1  |-  ( ( F  Fn  A  /\  X  e.  A )  ->  F  =  ( ( F  |`  ( A  \  { X } ) )  u.  { <. X ,  ( F `  X ) >. } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    \ cdif 3423    u. cun 3424   {csn 3975   <.cop 3981    |` cres 4940    Fn wfn 5511   ` cfv 5516
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524
This theorem is referenced by:  funresdfunsn  6019  ralxpmap  7362  finixpnum  28552
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