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Theorem dfimafn2 5564
Description: Alternate definition of the image of a function as an indexed union of singletons of function values. (Contributed by Raph Levien, 20-Nov-2006.)
Assertion
Ref Expression
dfimafn2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  U_ x  e.  A  { ( F `  x ) } )
Distinct variable groups:    x, A    x, F

Proof of Theorem dfimafn2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfimafn 5563 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  ( F `  x )  =  y } )
2 iunab 3932 . . 3  |-  U_ x  e.  A  { y  |  ( F `  x )  =  y }  =  { y  |  E. x  e.  A  ( F `  x )  =  y }
31, 2eqtr4di 2228 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  U_ x  e.  A  { y  |  ( F `  x )  =  y } )
4 df-sn 3598 . . . . 5  |-  { ( F `  x ) }  =  { y  |  y  =  ( F `  x ) }
5 eqcom 2179 . . . . . 6  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
65abbii 2293 . . . . 5  |-  { y  |  y  =  ( F `  x ) }  =  { y  |  ( F `  x )  =  y }
74, 6eqtri 2198 . . . 4  |-  { ( F `  x ) }  =  { y  |  ( F `  x )  =  y }
87a1i 9 . . 3  |-  ( x  e.  A  ->  { ( F `  x ) }  =  { y  |  ( F `  x )  =  y } )
98iuneq2i 3904 . 2  |-  U_ x  e.  A  { ( F `  x ) }  =  U_ x  e.  A  { y  |  ( F `  x
)  =  y }
103, 9eqtr4di 2228 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  U_ x  e.  A  { ( F `  x ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1353    e. wcel 2148   {cab 2163   E.wrex 2456    C_ wss 3129   {csn 3592   U_ciun 3886   dom cdm 4625   "cima 4628   Fun wfun 5209   ` cfv 5215
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4120  ax-pow 4173  ax-pr 4208
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4003  df-opab 4064  df-id 4292  df-xp 4631  df-rel 4632  df-cnv 4633  df-co 4634  df-dm 4635  df-rn 4636  df-res 4637  df-ima 4638  df-iota 5177  df-fun 5217  df-fn 5218  df-fv 5223
This theorem is referenced by: (None)
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