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Theorem unipreima 23899
Description: Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.)
Assertion
Ref Expression
unipreima  |-  ( Fun 
F  ->  ( `' F " U. A )  =  U_ x  e.  A  ( `' F " x ) )
Distinct variable groups:    x, F    x, A

Proof of Theorem unipreima
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 funfn 5423 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 r19.42v 2806 . . . . . . 7  |-  ( E. x  e.  A  ( y  e.  dom  F  /\  ( F `  y
)  e.  x )  <-> 
( y  e.  dom  F  /\  E. x  e.  A  ( F `  y )  e.  x
) )
32bicomi 194 . . . . . 6  |-  ( ( y  e.  dom  F  /\  E. x  e.  A  ( F `  y )  e.  x )  <->  E. x  e.  A  ( y  e.  dom  F  /\  ( F `  y )  e.  x ) )
43a1i 11 . . . . 5  |-  ( F  Fn  dom  F  -> 
( ( y  e. 
dom  F  /\  E. x  e.  A  ( F `  y )  e.  x
)  <->  E. x  e.  A  ( y  e.  dom  F  /\  ( F `  y )  e.  x
) ) )
5 eluni2 3962 . . . . . . 7  |-  ( ( F `  y )  e.  U. A  <->  E. x  e.  A  ( F `  y )  e.  x
)
65anbi2i 676 . . . . . 6  |-  ( ( y  e.  dom  F  /\  ( F `  y
)  e.  U. A
)  <->  ( y  e. 
dom  F  /\  E. x  e.  A  ( F `  y )  e.  x
) )
76a1i 11 . . . . 5  |-  ( F  Fn  dom  F  -> 
( ( y  e. 
dom  F  /\  ( F `  y )  e.  U. A )  <->  ( y  e.  dom  F  /\  E. x  e.  A  ( F `  y )  e.  x ) ) )
8 elpreima 5790 . . . . . 6  |-  ( F  Fn  dom  F  -> 
( y  e.  ( `' F " x )  <-> 
( y  e.  dom  F  /\  ( F `  y )  e.  x
) ) )
98rexbidv 2671 . . . . 5  |-  ( F  Fn  dom  F  -> 
( E. x  e.  A  y  e.  ( `' F " x )  <->  E. x  e.  A  ( y  e.  dom  F  /\  ( F `  y )  e.  x
) ) )
104, 7, 93bitr4d 277 . . . 4  |-  ( F  Fn  dom  F  -> 
( ( y  e. 
dom  F  /\  ( F `  y )  e.  U. A )  <->  E. x  e.  A  y  e.  ( `' F " x ) ) )
11 elpreima 5790 . . . 4  |-  ( F  Fn  dom  F  -> 
( y  e.  ( `' F " U. A
)  <->  ( y  e. 
dom  F  /\  ( F `  y )  e.  U. A ) ) )
12 eliun 4040 . . . . 5  |-  ( y  e.  U_ x  e.  A  ( `' F " x )  <->  E. x  e.  A  y  e.  ( `' F " x ) )
1312a1i 11 . . . 4  |-  ( F  Fn  dom  F  -> 
( y  e.  U_ x  e.  A  ( `' F " x )  <->  E. x  e.  A  y  e.  ( `' F " x ) ) )
1410, 11, 133bitr4d 277 . . 3  |-  ( F  Fn  dom  F  -> 
( y  e.  ( `' F " U. A
)  <->  y  e.  U_ x  e.  A  ( `' F " x ) ) )
1514eqrdv 2386 . 2  |-  ( F  Fn  dom  F  -> 
( `' F " U. A )  =  U_ x  e.  A  ( `' F " x ) )
161, 15sylbi 188 1  |-  ( Fun 
F  ->  ( `' F " U. A )  =  U_ x  e.  A  ( `' F " x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   E.wrex 2651   U.cuni 3958   U_ciun 4036   `'ccnv 4818   dom cdm 4819   "cima 4822   Fun wfun 5389    Fn wfn 5390   ` cfv 5395
This theorem is referenced by:  imambfm  24407  dstrvprob  24509
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-fv 5403
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