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Theorem fnunsn 5693
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 5640 . . . 4  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  { <. X ,  Y >. }  Fn  { X } )
52, 3, 4syl2anc 665 . . 3  |-  ( ph  ->  { <. X ,  Y >. }  Fn  { X } )
6 fnunop.d . . . 4  |-  ( ph  ->  -.  X  e.  D
)
7 disjsn 4054 . . . 4  |-  ( ( D  i^i  { X } )  =  (/)  <->  -.  X  e.  D )
86, 7sylibr 215 . . 3  |-  ( ph  ->  ( D  i^i  { X } )  =  (/) )
9 fnun 5692 . . 3  |-  ( ( ( F  Fn  D  /\  { <. X ,  Y >. }  Fn  { X } )  /\  ( D  i^i  { X }
)  =  (/) )  -> 
( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X }
) )
101, 5, 8, 9syl21anc 1263 . 2  |-  ( ph  ->  ( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X }
) )
11 fnunop.g . . . 4  |-  G  =  ( F  u.  { <. X ,  Y >. } )
1211fneq1i 5680 . . 3  |-  ( G  Fn  E  <->  ( F  u.  { <. X ,  Y >. } )  Fn  E
)
13 fnunop.e . . . 4  |-  E  =  ( D  u.  { X } )
1413fneq2i 5681 . . 3  |-  ( ( F  u.  { <. X ,  Y >. } )  Fn  E  <->  ( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X } ) )
1512, 14bitri 252 . 2  |-  ( G  Fn  E  <->  ( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X } ) )
1610, 15sylibr 215 1  |-  ( ph  ->  G  Fn  E )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1437    e. wcel 1867   _Vcvv 3078    u. cun 3431    i^i cin 3432   (/)c0 3758   {csn 3993   <.cop 3999    Fn wfn 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4477  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-fun 5595  df-fn 5596
This theorem is referenced by: (None)
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