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Theorem fsnunres 5721
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 5353 . . . 4  |-  ( F  Fn  S  ->  ( F  |`  S )  =  F )
21adantr 451 . . 3  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( F  |`  S )  =  F )
3 ressnop0 5703 . . . 4  |-  ( -.  X  e.  S  -> 
( { <. X ,  Y >. }  |`  S )  =  (/) )
43adantl 452 . . 3  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( { <. X ,  Y >. }  |`  S )  =  (/) )
52, 4uneq12d 3330 . 2  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( ( F  |`  S )  u.  ( { <. X ,  Y >. }  |`  S ) )  =  ( F  u.  (/) ) )
6 resundir 4970 . 2  |-  ( ( F  u.  { <. X ,  Y >. } )  |`  S )  =  ( ( F  |`  S )  u.  ( { <. X ,  Y >. }  |`  S ) )
7 un0 3479 . . 3  |-  ( F  u.  (/) )  =  F
87eqcomi 2287 . 2  |-  F  =  ( F  u.  (/) )
95, 6, 83eqtr4g 2340 1  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( ( F  u.  { <. X ,  Y >. } )  |`  S )  =  F )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684    u. cun 3150   (/)c0 3455   {csn 3640   <.cop 3643    |` cres 4691    Fn wfn 5250
This theorem is referenced by:  pgpfaclem1  15316  islindf4  27308
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  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-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-dm 4699  df-res 4701  df-fun 5257  df-fn 5258
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