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Theorem dfaimafn2 28134
Description: Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 5588. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
dfaimafn2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  U_ x  e.  A  { ( F''' x ) } )
Distinct variable groups:    x, A    x, F

Proof of Theorem dfaimafn2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dfaimafn 28133 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  ( F''' x )  =  y } )
2 iunab 3964 . . 3  |-  U_ x  e.  A  { y  |  ( F''' x )  =  y }  =  { y  |  E. x  e.  A  ( F''' x )  =  y }
31, 2syl6eqr 2346 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  U_ x  e.  A  { y  |  ( F''' x )  =  y } )
4 df-sn 3659 . . . . 5  |-  { ( F''' x ) }  =  { y  |  y  =  ( F''' x ) }
5 eqcom 2298 . . . . . 6  |-  ( y  =  ( F''' x )  <-> 
( F''' x )  =  y )
65abbii 2408 . . . . 5  |-  { y  |  y  =  ( F''' x ) }  =  { y  |  ( F''' x )  =  y }
74, 6eqtri 2316 . . . 4  |-  { ( F''' x ) }  =  { y  |  ( F''' x )  =  y }
87a1i 10 . . 3  |-  ( x  e.  A  ->  { ( F''' x ) }  =  { y  |  ( F''' x )  =  y } )
98iuneq2i 3939 . 2  |-  U_ x  e.  A  { ( F''' x ) }  =  U_ x  e.  A  {
y  |  ( F''' x )  =  y }
103, 9syl6eqr 2346 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 358    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557    C_ wss 3165   {csn 3653   U_ciun 3921   dom cdm 4705   "cima 4708   Fun wfun 5265  '''cafv 28075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279  df-dfat 28077  df-afv 28078
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