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Theorem fnimapr 5728
Description: The image of a pair under a funtion. (Contributed by Jeff Madsen, 6-Jan-2011.)
Assertion
Ref Expression
fnimapr  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  ( F " { B ,  C }
)  =  { ( F `  B ) ,  ( F `  C ) } )

Proof of Theorem fnimapr
StepHypRef Expression
1 fnsnfv 5727 . . . . 5  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `  B ) }  =  ( F " { B } ) )
213adant3 977 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  { ( F `  B ) }  =  ( F " { B } ) )
3 fnsnfv 5727 . . . . 5  |-  ( ( F  Fn  A  /\  C  e.  A )  ->  { ( F `  C ) }  =  ( F " { C } ) )
433adant2 976 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  { ( F `  C ) }  =  ( F " { C } ) )
52, 4uneq12d 3447 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  ( { ( F `
 B ) }  u.  { ( F `
 C ) } )  =  ( ( F " { B } )  u.  ( F " { C }
) ) )
65eqcomd 2394 . 2  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  ( ( F " { B } )  u.  ( F " { C } ) )  =  ( { ( F `
 B ) }  u.  { ( F `
 C ) } ) )
7 df-pr 3766 . . . 4  |-  { B ,  C }  =  ( { B }  u.  { C } )
87imaeq2i 5143 . . 3  |-  ( F
" { B ,  C } )  =  ( F " ( { B }  u.  { C } ) )
9 imaundi 5226 . . 3  |-  ( F
" ( { B }  u.  { C } ) )  =  ( ( F " { B } )  u.  ( F " { C } ) )
108, 9eqtri 2409 . 2  |-  ( F
" { B ,  C } )  =  ( ( F " { B } )  u.  ( F " { C }
) )
11 df-pr 3766 . 2  |-  { ( F `  B ) ,  ( F `  C ) }  =  ( { ( F `  B ) }  u.  { ( F `  C
) } )
126, 10, 113eqtr4g 2446 1  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  ( F " { B ,  C }
)  =  { ( F `  B ) ,  ( F `  C ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1649    e. wcel 1717    u. cun 3263   {csn 3759   {cpr 3760   "cima 4823    Fn wfn 5391   ` cfv 5396
This theorem is referenced by:  injresinjlem  11128  mrcun  13776  2pthonlem2  21450  constr3pthlem3  21494
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pr 4346
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-fv 5404
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