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Theorem fsnunres 6088
Description: Recover the original function from a point-added function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Assertion
Ref Expression
fsnunres  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( ( F  u.  { <. X ,  Y >. } )  |`  S )  =  F )

Proof of Theorem fsnunres
StepHypRef Expression
1 fnresdm 5672 . . . 4  |-  ( F  Fn  S  ->  ( F  |`  S )  =  F )
21adantr 463 . . 3  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( F  |`  S )  =  F )
3 ressnop0 6054 . . . 4  |-  ( -.  X  e.  S  -> 
( { <. X ,  Y >. }  |`  S )  =  (/) )
43adantl 464 . . 3  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( { <. X ,  Y >. }  |`  S )  =  (/) )
52, 4uneq12d 3645 . 2  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( ( F  |`  S )  u.  ( { <. X ,  Y >. }  |`  S ) )  =  ( F  u.  (/) ) )
6 resundir 5276 . 2  |-  ( ( F  u.  { <. X ,  Y >. } )  |`  S )  =  ( ( F  |`  S )  u.  ( { <. X ,  Y >. }  |`  S ) )
7 un0 3809 . . 3  |-  ( F  u.  (/) )  =  F
87eqcomi 2467 . 2  |-  F  =  ( F  u.  (/) )
95, 6, 83eqtr4g 2520 1  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( ( F  u.  { <. X ,  Y >. } )  |`  S )  =  F )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1398    e. wcel 1823    u. cun 3459   (/)c0 3783   {csn 4016   <.cop 4022    |` cres 4990    Fn wfn 5565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-dm 4998  df-res 5000  df-fun 5572  df-fn 5573
This theorem is referenced by:  pgpfaclem1  17327  islindf4  19040
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