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Theorem dfimafn2 5251
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 5250 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  ( F `  x )  =  y } )
2 iunab 3731 . . 3  |-  U_ x  e.  A  { y  |  ( F `  x )  =  y }  =  { y  |  E. x  e.  A  ( F `  x )  =  y }
31, 2syl6eqr 2106 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  U_ x  e.  A  { y  |  ( F `  x )  =  y } )
4 df-sn 3409 . . . . 5  |-  { ( F `  x ) }  =  { y  |  y  =  ( F `  x ) }
5 eqcom 2058 . . . . . 6  |-  ( y  =  ( F `  x )  <->  ( F `  x )  =  y )
65abbii 2169 . . . . 5  |-  { y  |  y  =  ( F `  x ) }  =  { y  |  ( F `  x )  =  y }
74, 6eqtri 2076 . . . 4  |-  { ( F `  x ) }  =  { y  |  ( F `  x )  =  y }
87a1i 9 . . 3  |-  ( x  e.  A  ->  { ( F `  x ) }  =  { y  |  ( F `  x )  =  y } )
98iuneq2i 3703 . 2  |-  U_ x  e.  A  { ( F `  x ) }  =  U_ x  e.  A  { y  |  ( F `  x
)  =  y }
103, 9syl6eqr 2106 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 101    = wceq 1259    e. wcel 1409   {cab 2042   E.wrex 2324    C_ wss 2945   {csn 3403   U_ciun 3685   dom cdm 4373   "cima 4376   Fun wfun 4924   ` cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-fv 4938
This theorem is referenced by: (None)
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