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Theorem fsnunres 6029
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 5629 . . . 4  |-  ( F  Fn  S  ->  ( F  |`  S )  =  F )
21adantr 465 . . 3  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( F  |`  S )  =  F )
3 ressnop0 5999 . . . 4  |-  ( -.  X  e.  S  -> 
( { <. X ,  Y >. }  |`  S )  =  (/) )
43adantl 466 . . 3  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( { <. X ,  Y >. }  |`  S )  =  (/) )
52, 4uneq12d 3620 . 2  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( ( F  |`  S )  u.  ( { <. X ,  Y >. }  |`  S ) )  =  ( F  u.  (/) ) )
6 resundir 5234 . 2  |-  ( ( F  u.  { <. X ,  Y >. } )  |`  S )  =  ( ( F  |`  S )  u.  ( { <. X ,  Y >. }  |`  S ) )
7 un0 3771 . . 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 369    = wceq 1370    e. wcel 1758    u. cun 3435   (/)c0 3746   {csn 3986   <.cop 3992    |` cres 4951    Fn wfn 5522
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 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640
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-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-br 4402  df-opab 4460  df-xp 4955  df-rel 4956  df-dm 4959  df-res 4961  df-fun 5529  df-fn 5530
This theorem is referenced by:  pgpfaclem1  16705  islindf4  18393
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