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Theorem fsnunf2 6086
Description: Adjoining a point to a punctured function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)
Assertion
Ref Expression
fsnunf2  |-  ( ( F : ( S 
\  { X }
) --> T  /\  X  e.  S  /\  Y  e.  T )  ->  ( F  u.  { <. X ,  Y >. } ) : S --> T )

Proof of Theorem fsnunf2
StepHypRef Expression
1 simp1 994 . . 3  |-  ( ( F : ( S 
\  { X }
) --> T  /\  X  e.  S  /\  Y  e.  T )  ->  F : ( S  \  { X } ) --> T )
2 simp2 995 . . 3  |-  ( ( F : ( S 
\  { X }
) --> T  /\  X  e.  S  /\  Y  e.  T )  ->  X  e.  S )
3 neldifsnd 4144 . . 3  |-  ( ( F : ( S 
\  { X }
) --> T  /\  X  e.  S  /\  Y  e.  T )  ->  -.  X  e.  ( S  \  { X } ) )
4 simp3 996 . . 3  |-  ( ( F : ( S 
\  { X }
) --> T  /\  X  e.  S  /\  Y  e.  T )  ->  Y  e.  T )
5 fsnunf 6085 . . 3  |-  ( ( F : ( S 
\  { X }
) --> T  /\  ( X  e.  S  /\  -.  X  e.  ( S  \  { X }
) )  /\  Y  e.  T )  ->  ( F  u.  { <. X ,  Y >. } ) : ( ( S  \  { X } )  u. 
{ X } ) --> T )
61, 2, 3, 4, 5syl121anc 1231 . 2  |-  ( ( F : ( S 
\  { X }
) --> T  /\  X  e.  S  /\  Y  e.  T )  ->  ( F  u.  { <. X ,  Y >. } ) : ( ( S  \  { X } )  u. 
{ X } ) --> T )
7 difsnid 4162 . . . 4  |-  ( X  e.  S  ->  (
( S  \  { X } )  u.  { X } )  =  S )
873ad2ant2 1016 . . 3  |-  ( ( F : ( S 
\  { X }
) --> T  /\  X  e.  S  /\  Y  e.  T )  ->  (
( S  \  { X } )  u.  { X } )  =  S )
98feq2d 5700 . 2  |-  ( ( F : ( S 
\  { X }
) --> T  /\  X  e.  S  /\  Y  e.  T )  ->  (
( F  u.  { <. X ,  Y >. } ) : ( ( S  \  { X } )  u.  { X } ) --> T  <->  ( F  u.  { <. X ,  Y >. } ) : S --> T ) )
106, 9mpbid 210 1  |-  ( ( F : ( S 
\  { X }
) --> T  /\  X  e.  S  /\  Y  e.  T )  ->  ( F  u.  { <. X ,  Y >. } ) : S --> T )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ w3a 971    = wceq 1398    e. wcel 1823    \ cdif 3458    u. cun 3459   {csn 4016   <.cop 4022   -->wf 5566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577
This theorem is referenced by:  fsets  14747  islindf4  19043
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