ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fsnunres Unicode version

Theorem fsnunres 5364
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 5008 . . . 4  |-  ( F  Fn  S  ->  ( F  |`  S )  =  F )
21adantr 261 . . 3  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( F  |`  S )  =  F )
3 ressnop0 5344 . . . 4  |-  ( -.  X  e.  S  -> 
( { <. X ,  Y >. }  |`  S )  =  (/) )
43adantl 262 . . 3  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( { <. X ,  Y >. }  |`  S )  =  (/) )
52, 4uneq12d 3098 . 2  |-  ( ( F  Fn  S  /\  -.  X  e.  S
)  ->  ( ( F  |`  S )  u.  ( { <. X ,  Y >. }  |`  S ) )  =  ( F  u.  (/) ) )
6 resundir 4626 . 2  |-  ( ( F  u.  { <. X ,  Y >. } )  |`  S )  =  ( ( F  |`  S )  u.  ( { <. X ,  Y >. }  |`  S ) )
7 un0 3251 . . 3  |-  ( F  u.  (/) )  =  F
87eqcomi 2044 . 2  |-  F  =  ( F  u.  (/) )
95, 6, 83eqtr4g 2097 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 97    = wceq 1243    e. wcel 1393    u. cun 2915   (/)c0 3224   {csn 3375   <.cop 3378    |` cres 4347    Fn wfn 4897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-dm 4355  df-res 4357  df-fun 4904  df-fn 4905
This theorem is referenced by:  tfrlemisucaccv  5939
  Copyright terms: Public domain W3C validator