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Theorem dfaimafn2 30069
Description: Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 5739. (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 30068 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  ( F''' x )  =  y } )
2 iunab 4214 . . 3  |-  U_ x  e.  A  { y  |  ( F''' x )  =  y }  =  { y  |  E. x  e.  A  ( F''' x )  =  y }
31, 2syl6eqr 2491 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  U_ x  e.  A  { y  |  ( F''' x )  =  y } )
4 df-sn 3876 . . . . 5  |-  { ( F''' x ) }  =  { y  |  y  =  ( F''' x ) }
5 eqcom 2443 . . . . . 6  |-  ( y  =  ( F''' x )  <-> 
( F''' x )  =  y )
65abbii 2553 . . . . 5  |-  { y  |  y  =  ( F''' x ) }  =  { y  |  ( F''' x )  =  y }
74, 6eqtri 2461 . . . 4  |-  { ( F''' x ) }  =  { y  |  ( F''' x )  =  y }
87a1i 11 . . 3  |-  ( x  e.  A  ->  { ( F''' x ) }  =  { y  |  ( F''' x )  =  y } )
98iuneq2i 4187 . 2  |-  U_ x  e.  A  { ( F''' x ) }  =  U_ x  e.  A  {
y  |  ( F''' x )  =  y }
103, 9syl6eqr 2491 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  U_ x  e.  A  { ( F''' x ) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   {cab 2427   E.wrex 2714    C_ wss 3326   {csn 3875   U_ciun 4169   dom cdm 4838   "cima 4841   Fun wfun 5410  '''cafv 30015
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3185  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-iun 4171  df-br 4291  df-opab 4349  df-id 4634  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-fv 5424  df-dfat 30017  df-afv 30018
This theorem is referenced by: (None)
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