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Theorem dfaimafn2 28007
Description: Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 5777. (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 28006 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  ( F''' x )  =  y } )
2 iunab 4138 . . 3  |-  U_ x  e.  A  { y  |  ( F''' x )  =  y }  =  { y  |  E. x  e.  A  ( F''' x )  =  y }
31, 2syl6eqr 2487 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  U_ x  e.  A  { y  |  ( F''' x )  =  y } )
4 df-sn 3821 . . . . 5  |-  { ( F''' x ) }  =  { y  |  y  =  ( F''' x ) }
5 eqcom 2439 . . . . . 6  |-  ( y  =  ( F''' x )  <-> 
( F''' x )  =  y )
65abbii 2549 . . . . 5  |-  { y  |  y  =  ( F''' x ) }  =  { y  |  ( F''' x )  =  y }
74, 6eqtri 2457 . . . 4  |-  { ( F''' x ) }  =  { y  |  ( F''' x )  =  y }
87a1i 11 . . 3  |-  ( x  e.  A  ->  { ( F''' x ) }  =  { y  |  ( F''' x )  =  y } )
98iuneq2i 4112 . 2  |-  U_ x  e.  A  { ( F''' x ) }  =  U_ x  e.  A  {
y  |  ( F''' x )  =  y }
103, 9syl6eqr 2487 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 360    = wceq 1653    e. wcel 1726   {cab 2423   E.wrex 2707    C_ wss 3321   {csn 3815   U_ciun 4094   dom cdm 4879   "cima 4882   Fun wfun 5449  '''cafv 27949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-fv 5463  df-dfat 27951  df-afv 27952
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