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Theorem fnimapr 5621
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 5620 . . . . 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 5620 . . . . 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 3364 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  ( { ( F `
 B ) }  u.  { ( F `
 C ) } )  =  ( ( F " { B } )  u.  ( F " { C }
) ) )
65eqcomd 2321 . 2  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  ( ( F " { B } )  u.  ( F " { C } ) )  =  ( { ( F `
 B ) }  u.  { ( F `
 C ) } ) )
7 df-pr 3681 . . . 4  |-  { B ,  C }  =  ( { B }  u.  { C } )
87imaeq2i 5047 . . 3  |-  ( F
" { B ,  C } )  =  ( F " ( { B }  u.  { C } ) )
9 imaundi 5130 . . 3  |-  ( F
" ( { B }  u.  { C } ) )  =  ( ( F " { B } )  u.  ( F " { C } ) )
108, 9eqtri 2336 . 2  |-  ( F
" { B ,  C } )  =  ( ( F " { B } )  u.  ( F " { C }
) )
11 df-pr 3681 . 2  |-  { ( F `  B ) ,  ( F `  C ) }  =  ( { ( F `  B ) }  u.  { ( F `  C
) } )
126, 10, 113eqtr4g 2373 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 1633    e. wcel 1701    u. cun 3184   {csn 3674   {cpr 3675   "cima 4729    Fn wfn 5287   ` cfv 5292
This theorem is referenced by:  mrcun  13573  fnimaprOLD  25521  injresinjlem  27276  2pthonlem2  27496  constr3pthlem3  27541
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-fv 5300
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