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Theorem fnimapr 5583
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 5582 . . . . 5  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `  B ) }  =  ( F " { B } ) )
213adant3 975 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  { ( F `  B ) }  =  ( F " { B } ) )
3 fnsnfv 5582 . . . . 5  |-  ( ( F  Fn  A  /\  C  e.  A )  ->  { ( F `  C ) }  =  ( F " { C } ) )
433adant2 974 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  { ( F `  C ) }  =  ( F " { C } ) )
52, 4uneq12d 3330 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  ( { ( F `
 B ) }  u.  { ( F `
 C ) } )  =  ( ( F " { B } )  u.  ( F " { C }
) ) )
65eqcomd 2288 . 2  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  ( ( F " { B } )  u.  ( F " { C } ) )  =  ( { ( F `
 B ) }  u.  { ( F `
 C ) } ) )
7 df-pr 3647 . . . 4  |-  { B ,  C }  =  ( { B }  u.  { C } )
87imaeq2i 5010 . . 3  |-  ( F
" { B ,  C } )  =  ( F " ( { B }  u.  { C } ) )
9 imaundi 5093 . . 3  |-  ( F
" ( { B }  u.  { C } ) )  =  ( ( F " { B } )  u.  ( F " { C } ) )
108, 9eqtri 2303 . 2  |-  ( F
" { B ,  C } )  =  ( ( F " { B } )  u.  ( F " { C }
) )
11 df-pr 3647 . 2  |-  { ( F `  B ) ,  ( F `  C ) }  =  ( { ( F `  B ) }  u.  { ( F `  C
) } )
126, 10, 113eqtr4g 2340 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 934    = wceq 1623    e. wcel 1684    u. cun 3150   {csn 3640   {cpr 3641   "cima 4692    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  mrcun  13524  fnimaprOLD  25784
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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