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Theorem dfimafn2 5338
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 5337 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  ( F `  x )  =  y } )
2 iunab 3771 . . 3  |-  U_ x  e.  A  { y  |  ( F `  x )  =  y }  =  { y  |  E. x  e.  A  ( F `  x )  =  y }
31, 2syl6eqr 2138 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  U_ x  e.  A  { y  |  ( F `  x )  =  y } )
4 df-sn 3447 . . . . 5  |-  { ( F `  x ) }  =  { y  |  y  =  ( F `  x ) }
5 eqcom 2090 . . . . . 6  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
65abbii 2203 . . . . 5  |-  { y  |  y  =  ( F `  x ) }  =  { y  |  ( F `  x )  =  y }
74, 6eqtri 2108 . . . 4  |-  { ( F `  x ) }  =  { y  |  ( F `  x )  =  y }
87a1i 9 . . 3  |-  ( x  e.  A  ->  { ( F `  x ) }  =  { y  |  ( F `  x )  =  y } )
98iuneq2i 3743 . 2  |-  U_ x  e.  A  { ( F `  x ) }  =  U_ x  e.  A  { y  |  ( F `  x
)  =  y }
103, 9syl6eqr 2138 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 102    = wceq 1289    e. wcel 1438   {cab 2074   E.wrex 2360    C_ wss 2997   {csn 3441   U_ciun 3725   dom cdm 4428   "cima 4431   Fun wfun 4996   ` cfv 5002
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2839  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-fv 5010
This theorem is referenced by: (None)
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