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Theorem fnimapr 5265
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 5264 . . . . 5  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  { ( F `  B ) }  =  ( F " { B } ) )
213adant3 959 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  { ( F `  B ) }  =  ( F " { B } ) )
3 fnsnfv 5264 . . . . 5  |-  ( ( F  Fn  A  /\  C  e.  A )  ->  { ( F `  C ) }  =  ( F " { C } ) )
433adant2 958 . . . 4  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  { ( F `  C ) }  =  ( F " { C } ) )
52, 4uneq12d 3128 . . 3  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  ( { ( F `
 B ) }  u.  { ( F `
 C ) } )  =  ( ( F " { B } )  u.  ( F " { C }
) ) )
65eqcomd 2087 . 2  |-  ( ( F  Fn  A  /\  B  e.  A  /\  C  e.  A )  ->  ( ( F " { B } )  u.  ( F " { C } ) )  =  ( { ( F `
 B ) }  u.  { ( F `
 C ) } ) )
7 df-pr 3413 . . . 4  |-  { B ,  C }  =  ( { B }  u.  { C } )
87imaeq2i 4696 . . 3  |-  ( F
" { B ,  C } )  =  ( F " ( { B }  u.  { C } ) )
9 imaundi 4766 . . 3  |-  ( F
" ( { B }  u.  { C } ) )  =  ( ( F " { B } )  u.  ( F " { C } ) )
108, 9eqtri 2102 . 2  |-  ( F
" { B ,  C } )  =  ( ( F " { B } )  u.  ( F " { C }
) )
11 df-pr 3413 . 2  |-  { ( F `  B ) ,  ( F `  C ) }  =  ( { ( F `  B ) }  u.  { ( F `  C
) } )
126, 10, 113eqtr4g 2139 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 920    = wceq 1285    e. wcel 1434    u. cun 2972   {csn 3406   {cpr 3407   "cima 4374    Fn wfn 4927   ` cfv 4932
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-sbc 2817  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-br 3794  df-opab 3848  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-fv 4940
This theorem is referenced by:  fvinim0ffz  9327
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