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Theorem fnunsn 5361
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 5301 . . . 4  |-  ( ( X  e.  _V  /\  Y  e.  _V )  ->  { <. X ,  Y >. }  Fn  { X } )
52, 3, 4syl2anc 411 . . 3  |-  ( ph  ->  { <. X ,  Y >. }  Fn  { X } )
6 fnunop.d . . . 4  |-  ( ph  ->  -.  X  e.  D
)
7 disjsn 3680 . . . 4  |-  ( ( D  i^i  { X } )  =  (/)  <->  -.  X  e.  D )
86, 7sylibr 134 . . 3  |-  ( ph  ->  ( D  i^i  { X } )  =  (/) )
9 fnun 5360 . . 3  |-  ( ( ( F  Fn  D  /\  { <. X ,  Y >. }  Fn  { X } )  /\  ( D  i^i  { X }
)  =  (/) )  -> 
( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X }
) )
101, 5, 8, 9syl21anc 1248 . 2  |-  ( ph  ->  ( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X }
) )
11 fnunop.g . . . 4  |-  G  =  ( F  u.  { <. X ,  Y >. } )
1211fneq1i 5348 . . 3  |-  ( G  Fn  E  <->  ( F  u.  { <. X ,  Y >. } )  Fn  E
)
13 fnunop.e . . . 4  |-  E  =  ( D  u.  { X } )
1413fneq2i 5349 . . 3  |-  ( ( F  u.  { <. X ,  Y >. } )  Fn  E  <->  ( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X } ) )
1512, 14bitri 184 . 2  |-  ( G  Fn  E  <->  ( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X } ) )
1610, 15sylibr 134 1  |-  ( ph  ->  G  Fn  E )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1364    e. wcel 2164   _Vcvv 2760    u. cun 3151    i^i cin 3152   (/)c0 3446   {csn 3618   <.cop 3621    Fn wfn 5249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-fun 5256  df-fn 5257
This theorem is referenced by:  tfrlemisucfn  6377  tfr1onlemsucfn  6393
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