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Theorem fsnunres 5737
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 5369 . . . 4  |-  ( F  Fn  S  ->  ( F  |`  S )  =  F )
21adantr 451 . . 3  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( F  |`  S )  =  F )
3 ressnop0 5719 . . . 4  |-  ( -.  X  e.  S  -> 
( { <. X ,  Y >. }  |`  S )  =  (/) )
43adantl 452 . . 3  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( { <. X ,  Y >. }  |`  S )  =  (/) )
52, 4uneq12d 3343 . 2  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( ( F  |`  S )  u.  ( { <. X ,  Y >. }  |`  S ) )  =  ( F  u.  (/) ) )
6 resundir 4986 . 2  |-  ( ( F  u.  { <. X ,  Y >. } )  |`  S )  =  ( ( F  |`  S )  u.  ( { <. X ,  Y >. }  |`  S ) )
7 un0 3492 . . 3  |-  ( F  u.  (/) )  =  F
87eqcomi 2300 . 2  |-  F  =  ( F  u.  (/) )
95, 6, 83eqtr4g 2353 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 1632    e. wcel 1696    u. cun 3163   (/)c0 3468   {csn 3653   <.cop 3656    |` cres 4707    Fn wfn 5266
This theorem is referenced by:  pgpfaclem1  15332  islindf4  27411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-dm 4715  df-res 4717  df-fun 5273  df-fn 5274
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