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Theorem dfaimafn2 32412
Description: Alternate definition of the image of a function as an indexed union of singletons of function values, analogous to dfimafn2 5923. (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 32411 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  { y  |  E. x  e.  A  ( F''' x )  =  y } )
2 iunab 4378 . . 3  |-  U_ x  e.  A  { y  |  ( F''' x )  =  y }  =  { y  |  E. x  e.  A  ( F''' x )  =  y }
31, 2syl6eqr 2516 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( F " A
)  =  U_ x  e.  A  { y  |  ( F''' x )  =  y } )
4 df-sn 4033 . . . . 5  |-  { ( F''' x ) }  =  { y  |  y  =  ( F''' x ) }
5 eqcom 2466 . . . . . 6  |-  ( y  =  ( F''' x )  <-> 
( F''' x )  =  y )
65abbii 2591 . . . . 5  |-  { y  |  y  =  ( F''' x ) }  =  { y  |  ( F''' x )  =  y }
74, 6eqtri 2486 . . . 4  |-  { ( F''' x ) }  =  { y  |  ( F''' x )  =  y }
87a1i 11 . . 3  |-  ( x  e.  A  ->  { ( F''' x ) }  =  { y  |  ( F''' x )  =  y } )
98iuneq2i 4351 . 2  |-  U_ x  e.  A  { ( F''' x ) }  =  U_ x  e.  A  {
y  |  ( F''' x )  =  y }
103, 9syl6eqr 2516 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 1395    e. wcel 1819   {cab 2442   E.wrex 2808    C_ wss 3471   {csn 4032   U_ciun 4332   dom cdm 5008   "cima 5011   Fun wfun 5588  '''cafv 32360
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602  df-dfat 32362  df-afv 32363
This theorem is referenced by: (None)
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