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Theorem fdmdifeqresdif 32226
Description: The restriction of a conditional mapping to function values of a function having a domain which is a difference with a singleton equals this function. (Contributed by AV, 23-Apr-2019.)
Hypothesis
Ref Expression
fdmdifeqresdif.f  |-  F  =  ( x  e.  D  |->  if ( x  =  Y ,  X , 
( G `  x
) ) )
Assertion
Ref Expression
fdmdifeqresdif  |-  ( G : ( D  \  { Y } ) --> R  ->  G  =  ( F  |`  ( D  \  { Y } ) ) )
Distinct variable groups:    x, D    x, G    x, R    x, Y
Allowed substitution hints:    F( x)    X( x)

Proof of Theorem fdmdifeqresdif
StepHypRef Expression
1 eldifsn 4152 . . . . . 6  |-  ( x  e.  ( D  \  { Y } )  <->  ( x  e.  D  /\  x  =/=  Y ) )
2 df-ne 2664 . . . . . . . 8  |-  ( x  =/=  Y  <->  -.  x  =  Y )
32biimpi 194 . . . . . . 7  |-  ( x  =/=  Y  ->  -.  x  =  Y )
43adantl 466 . . . . . 6  |-  ( ( x  e.  D  /\  x  =/=  Y )  ->  -.  x  =  Y
)
51, 4sylbi 195 . . . . 5  |-  ( x  e.  ( D  \  { Y } )  ->  -.  x  =  Y
)
65adantl 466 . . . 4  |-  ( ( G : ( D 
\  { Y }
) --> R  /\  x  e.  ( D  \  { Y } ) )  ->  -.  x  =  Y
)
7 iffalse 3948 . . . 4  |-  ( -.  x  =  Y  ->  if ( x  =  Y ,  X ,  ( G `  x ) )  =  ( G `
 x ) )
86, 7syl 16 . . 3  |-  ( ( G : ( D 
\  { Y }
) --> R  /\  x  e.  ( D  \  { Y } ) )  ->  if ( x  =  Y ,  X ,  ( G `  x ) )  =  ( G `
 x ) )
98mpteq2dva 4533 . 2  |-  ( G : ( D  \  { Y } ) --> R  ->  ( x  e.  ( D  \  { Y } )  |->  if ( x  =  Y ,  X ,  ( G `  x ) ) )  =  ( x  e.  ( D  \  { Y } )  |->  ( G `
 x ) ) )
10 fdmdifeqresdif.f . . . 4  |-  F  =  ( x  e.  D  |->  if ( x  =  Y ,  X , 
( G `  x
) ) )
1110reseq1i 5269 . . 3  |-  ( F  |`  ( D  \  { Y } ) )  =  ( ( x  e.  D  |->  if ( x  =  Y ,  X ,  ( G `  x ) ) )  |`  ( D  \  { Y } ) )
12 difssd 3632 . . . 4  |-  ( G : ( D  \  { Y } ) --> R  ->  ( D  \  { Y } )  C_  D )
13 resmpt 5323 . . . 4  |-  ( ( D  \  { Y } )  C_  D  ->  ( ( x  e.  D  |->  if ( x  =  Y ,  X ,  ( G `  x ) ) )  |`  ( D  \  { Y } ) )  =  ( x  e.  ( D  \  { Y } )  |->  if ( x  =  Y ,  X ,  ( G `  x ) ) ) )
1412, 13syl 16 . . 3  |-  ( G : ( D  \  { Y } ) --> R  ->  ( ( x  e.  D  |->  if ( x  =  Y ,  X ,  ( G `  x ) ) )  |`  ( D  \  { Y } ) )  =  ( x  e.  ( D  \  { Y } )  |->  if ( x  =  Y ,  X ,  ( G `  x ) ) ) )
1511, 14syl5eq 2520 . 2  |-  ( G : ( D  \  { Y } ) --> R  ->  ( F  |`  ( D  \  { Y } ) )  =  ( x  e.  ( D  \  { Y } )  |->  if ( x  =  Y ,  X ,  ( G `  x ) ) ) )
16 ffn 5731 . . 3  |-  ( G : ( D  \  { Y } ) --> R  ->  G  Fn  ( D  \  { Y }
) )
17 dffn5 5913 . . 3  |-  ( G  Fn  ( D  \  { Y } )  <->  G  =  ( x  e.  ( D  \  { Y }
)  |->  ( G `  x ) ) )
1816, 17sylib 196 . 2  |-  ( G : ( D  \  { Y } ) --> R  ->  G  =  ( x  e.  ( D 
\  { Y }
)  |->  ( G `  x ) ) )
199, 15, 183eqtr4rd 2519 1  |-  ( G : ( D  \  { Y } ) --> R  ->  G  =  ( F  |`  ( D  \  { Y } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    \ cdif 3473    C_ wss 3476   ifcif 3939   {csn 4027    |-> cmpt 4505    |` cres 5001    Fn wfn 5583   -->wf 5584   ` cfv 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596
This theorem is referenced by:  lincext2  32354  lincext3  32355
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