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Theorem fnunsn 5581
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 5527 . . . 4  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  { <. X ,  Y >. }  Fn  { X } )
52, 3, 4syl2anc 644 . . 3  |-  ( ph  ->  { <. X ,  Y >. }  Fn  { X } )
6 fnunop.d . . . 4  |-  ( ph  ->  -.  X  e.  D
)
7 disjsn 3892 . . . 4  |-  ( ( D  i^i  { X } )  =  (/)  <->  -.  X  e.  D )
86, 7sylibr 205 . . 3  |-  ( ph  ->  ( D  i^i  { X } )  =  (/) )
9 fnun 5580 . . 3  |-  ( ( ( F  Fn  D  /\  { <. X ,  Y >. }  Fn  { X } )  /\  ( D  i^i  { X }
)  =  (/) )  -> 
( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X }
) )
101, 5, 8, 9syl21anc 1184 . 2  |-  ( ph  ->  ( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X }
) )
11 fnunop.g . . . 4  |-  G  =  ( F  u.  { <. X ,  Y >. } )
1211fneq1i 5568 . . 3  |-  ( G  Fn  E  <->  ( F  u.  { <. X ,  Y >. } )  Fn  E
)
13 fnunop.e . . . 4  |-  E  =  ( D  u.  { X } )
1413fneq2i 5569 . . 3  |-  ( ( F  u.  { <. X ,  Y >. } )  Fn  E  <->  ( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X } ) )
1512, 14bitri 242 . 2  |-  ( G  Fn  E  <->  ( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X } ) )
1610, 15sylibr 205 1  |-  ( ph  ->  G  Fn  E )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1653    e. wcel 1727   _Vcvv 2962    u. cun 3304    i^i cin 3305   (/)c0 3613   {csn 3838   <.cop 3841    Fn wfn 5478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-br 4238  df-opab 4292  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-fun 5485  df-fn 5486
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