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Theorem fnunsn 5319
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 5259 . . . 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 3653 . . . 4  |-  ( ( D  i^i  { X } )  =  (/)  <->  -.  X  e.  D )
86, 7sylibr 134 . . 3  |-  ( ph  ->  ( D  i^i  { X } )  =  (/) )
9 fnun 5318 . . 3  |-  ( ( ( F  Fn  D  /\  { <. X ,  Y >. }  Fn  { X } )  /\  ( D  i^i  { X }
)  =  (/) )  -> 
( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X }
) )
101, 5, 8, 9syl21anc 1237 . 2  |-  ( ph  ->  ( F  u.  { <. X ,  Y >. } )  Fn  ( D  u.  { X }
) )
11 fnunop.g . . . 4  |-  G  =  ( F  u.  { <. X ,  Y >. } )
1211fneq1i 5306 . . 3  |-  ( G  Fn  E  <->  ( F  u.  { <. X ,  Y >. } )  Fn  E
)
13 fnunop.e . . . 4  |-  E  =  ( D  u.  { X } )
1413fneq2i 5307 . . 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 1353    e. wcel 2148   _Vcvv 2737    u. cun 3127    i^i cin 3128   (/)c0 3422   {csn 3591   <.cop 3594    Fn wfn 5207
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-fun 5214  df-fn 5215
This theorem is referenced by:  tfrlemisucfn  6319  tfr1onlemsucfn  6335
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