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Theorem fnimapr 5857
Description: The image of a pair under a function. (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 5853 . . . . 5  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `  B ) }  =  ( F " { B } ) )
213adant3 1008 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  { ( F `  B ) }  =  ( F " { B } ) )
3 fnsnfv 5853 . . . . 5  |-  ( ( F  Fn  A  /\  C  e.  A )  ->  { ( F `  C ) }  =  ( F " { C } ) )
433adant2 1007 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  { ( F `  C ) }  =  ( F " { C } ) )
52, 4uneq12d 3612 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  ( { ( F `
 B ) }  u.  { ( F `
 C ) } )  =  ( ( F " { B } )  u.  ( F " { C }
) ) )
65eqcomd 2459 . 2  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  ( ( F " { B } )  u.  ( F " { C } ) )  =  ( { ( F `
 B ) }  u.  { ( F `
 C ) } ) )
7 df-pr 3981 . . . 4  |-  { B ,  C }  =  ( { B }  u.  { C } )
87imaeq2i 5268 . . 3  |-  ( F
" { B ,  C } )  =  ( F " ( { B }  u.  { C } ) )
9 imaundi 5350 . . 3  |-  ( F
" ( { B }  u.  { C } ) )  =  ( ( F " { B } )  u.  ( F " { C } ) )
108, 9eqtri 2480 . 2  |-  ( F
" { B ,  C } )  =  ( ( F " { B } )  u.  ( F " { C }
) )
11 df-pr 3981 . 2  |-  { ( F `  B ) ,  ( F `  C ) }  =  ( { ( F `  B ) }  u.  { ( F `  C
) } )
126, 10, 113eqtr4g 2517 1  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  ( F " { B ,  C }
)  =  { ( F `  B ) ,  ( F `  C ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 965    = wceq 1370    e. wcel 1758    u. cun 3427   {csn 3978   {cpr 3980   "cima 4944    Fn wfn 5514   ` cfv 5519
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-fv 5527
This theorem is referenced by:  injresinjlem  11748  mrcun  14671  2pthlem2  23640  constr3pthlem3  23688  imarnf1pr  30291  usgra2adedglem1  30449
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