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Theorem unipreima 28247
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 5630 . 2  |-  ( Fun 
F  <->  F  Fn  dom  F )
2 r19.42v 2980 . . . . . . 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 205 . . . . . 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 4223 . . . . . . 7  |-  ( ( F `  y )  e.  U. A  <->  E. x  e.  A  ( F `  y )  e.  x
)
65anbi2i 698 . . . . . 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 6017 . . . . . 6  |-  ( F  Fn  dom  F  -> 
( y  e.  ( `' F " x )  <-> 
( y  e.  dom  F  /\  ( F `  y )  e.  x
) ) )
98rexbidv 2936 . . . . 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 288 . . . 4  |-  ( F  Fn  dom  F  -> 
( ( y  e. 
dom  F  /\  ( F `  y )  e.  U. A )  <->  E. x  e.  A  y  e.  ( `' F " x ) ) )
11 elpreima 6017 . . . 4  |-  ( F  Fn  dom  F  -> 
( y  e.  ( `' F " U. A
)  <->  ( y  e. 
dom  F  /\  ( F `  y )  e.  U. A ) ) )
12 eliun 4304 . . . . 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 288 . . 3  |-  ( F  Fn  dom  F  -> 
( y  e.  ( `' F " U. A
)  <->  y  e.  U_ x  e.  A  ( `' F " x ) ) )
1514eqrdv 2419 . 2  |-  ( F  Fn  dom  F  -> 
( `' F " U. A )  =  U_ x  e.  A  ( `' F " x ) )
161, 15sylbi 198 1  |-  ( Fun 
F  ->  ( `' F " U. A )  =  U_ x  e.  A  ( `' F " x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1872   E.wrex 2772   U.cuni 4219   U_ciun 4299   `'ccnv 4852   dom cdm 4853   "cima 4856   Fun wfun 5595    Fn wfn 5596   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609
This theorem is referenced by:  imambfm  29092  dstrvprob  29312
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