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Theorem unipreima 24048
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 5474 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 r19.42v 2854 . . . . . . 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 4011 . . . . . . 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 5842 . . . . . 6  |-  ( F  Fn  dom  F  -> 
( y  e.  ( `' F " x )  <-> 
( y  e.  dom  F  /\  ( F `  y )  e.  x
) ) )
98rexbidv 2718 . . . . 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 5842 . . . 4  |-  ( F  Fn  dom  F  -> 
( y  e.  ( `' F " U. A
)  <->  ( y  e. 
dom  F  /\  ( F `  y )  e.  U. A ) ) )
12 eliun 4089 . . . . 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 2433 . 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 1652    e. wcel 1725   E.wrex 2698   U.cuni 4007   U_ciun 4085   `'ccnv 4869   dom cdm 4870   "cima 4873   Fun wfun 5440    Fn wfn 5441   ` cfv 5446
This theorem is referenced by:  imambfm  24604  dstrvprob  24721
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-fv 5454
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