MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fnunsn Unicode version

Theorem fnunsn 5351
Description: Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)
Hypotheses
Ref Expression
fnunop.x  |-  ( ph  ->  X  e.  _V )
fnunop.y  |-  ( ph  ->  Y  e.  _V )
fnunop.f  |-  ( ph  ->  F  Fn  D )
fnunop.g  |-  G  =  ( F  u.  { <. X ,  Y >. } )
fnunop.e  |-  E  =  ( D  u.  { X } )
fnunop.d  |-  ( ph  ->  -.  X  e.  D
)
Assertion
Ref Expression
fnunsn  |-  ( ph  ->  G  Fn  E )

Proof of Theorem fnunsn
StepHypRef Expression
1 fnunop.f . . 3  |-  ( ph  ->  F  Fn  D )
2 fnunop.x . . . 4  |-  ( ph  ->  X  e.  _V )
3 fnunop.y . . . 4  |-  ( ph  ->  Y  e.  _V )
4 fnsng 5299 . . . 4  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  { <. X ,  Y >. }  Fn  { X } )
52, 3, 4syl2anc 642 . . 3  |-  ( ph  ->  { <. X ,  Y >. }  Fn  { X } )
6 fnunop.d . . . 4  |-  ( ph  ->  -.  X  e.  D
)
7 disjsn 3693 . . . 4  |-  ( ( D  i^i  { X } )  =  (/)  <->  -.  X  e.  D )
86, 7sylibr 203 . . 3  |-  ( ph  ->  ( D  i^i  { X } )  =  (/) )
9 fnun 5350 . . 3  |-  ( ( ( F  Fn  D  /\  { <. X ,  Y >. }  Fn  { X } )  /\  ( D  i^i  { X }
)  =  (/) )  -> 
( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X }
) )
101, 5, 8, 9syl21anc 1181 . 2  |-  ( ph  ->  ( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X }
) )
11 fnunop.g . . . 4  |-  G  =  ( F  u.  { <. X ,  Y >. } )
1211fneq1i 5338 . . 3  |-  ( G  Fn  E  <->  ( F  u.  { <. X ,  Y >. } )  Fn  E
)
13 fnunop.e . . . 4  |-  E  =  ( D  u.  { X } )
1413fneq2i 5339 . . 3  |-  ( ( F  u.  { <. X ,  Y >. } )  Fn  E  <->  ( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X } ) )
1512, 14bitri 240 . 2  |-  ( G  Fn  E  <->  ( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X } ) )
1610, 15sylibr 203 1  |-  ( ph  ->  G  Fn  E )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788    u. cun 3150    i^i cin 3151   (/)c0 3455   {csn 3640   <.cop 3643    Fn wfn 5250
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-eu 2147  df-mo 2148  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-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257  df-fn 5258
  Copyright terms: Public domain W3C validator