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Theorem dfimafn2 5223
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 5222 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  ( F `  x )  =  y } )
2 iunab 3703 . . 3  |-  U_ x  e.  A  { y  |  ( F `  x )  =  y }  =  { y  |  E. x  e.  A  ( F `  x )  =  y }
31, 2syl6eqr 2090 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  U_ x  e.  A  { y  |  ( F `  x )  =  y } )
4 df-sn 3381 . . . . 5  |-  { ( F `  x ) }  =  { y  |  y  =  ( F `  x ) }
5 eqcom 2042 . . . . . 6  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
65abbii 2153 . . . . 5  |-  { y  |  y  =  ( F `  x ) }  =  { y  |  ( F `  x )  =  y }
74, 6eqtri 2060 . . . 4  |-  { ( F `  x ) }  =  { y  |  ( F `  x )  =  y }
87a1i 9 . . 3  |-  ( x  e.  A  ->  { ( F `  x ) }  =  { y  |  ( F `  x )  =  y } )
98iuneq2i 3675 . 2  |-  U_ x  e.  A  { ( F `  x ) }  =  U_ x  e.  A  { y  |  ( F `  x
)  =  y }
103, 9syl6eqr 2090 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 97    = wceq 1243    e. wcel 1393   {cab 2026   E.wrex 2307    C_ wss 2917   {csn 3375   U_ciun 3657   dom cdm 4345   "cima 4348   Fun wfun 4896   ` cfv 4902
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910
This theorem is referenced by: (None)
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