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Theorem unipreima 27913
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 5597 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 r19.42v 2961 . . . . . . 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 202 . . . . . 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 4194 . . . . . . 7  |-  ( ( F `  y )  e.  U. A  <->  E. x  e.  A  ( F `  y )  e.  x
)
65anbi2i 692 . . . . . 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 5984 . . . . . 6  |-  ( F  Fn  dom  F  -> 
( y  e.  ( `' F " x )  <-> 
( y  e.  dom  F  /\  ( F `  y )  e.  x
) ) )
98rexbidv 2917 . . . . 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 285 . . . 4  |-  ( F  Fn  dom  F  -> 
( ( y  e. 
dom  F  /\  ( F `  y )  e.  U. A )  <->  E. x  e.  A  y  e.  ( `' F " x ) ) )
11 elpreima 5984 . . . 4  |-  ( F  Fn  dom  F  -> 
( y  e.  ( `' F " U. A
)  <->  ( y  e. 
dom  F  /\  ( F `  y )  e.  U. A ) ) )
12 eliun 4275 . . . . 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 285 . . 3  |-  ( F  Fn  dom  F  -> 
( y  e.  ( `' F " U. A
)  <->  y  e.  U_ x  e.  A  ( `' F " x ) ) )
1514eqrdv 2399 . 2  |-  ( F  Fn  dom  F  -> 
( `' F " U. A )  =  U_ x  e.  A  ( `' F " x ) )
161, 15sylbi 195 1  |-  ( Fun 
F  ->  ( `' F " U. A )  =  U_ x  e.  A  ( `' F " x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   E.wrex 2754   U.cuni 4190   U_ciun 4270   `'ccnv 4821   dom cdm 4822   "cima 4825   Fun wfun 5562    Fn wfn 5563   ` cfv 5568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-rab 2762  df-v 3060  df-sbc 3277  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-iun 4272  df-br 4395  df-opab 4453  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-fv 5576
This theorem is referenced by:  imambfm  28696  dstrvprob  28902
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