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Theorem iunpreima 28170
Description: Preimage of an indexed union. (Contributed by Thierry Arnoux, 27-Mar-2018.)
Assertion
Ref Expression
iunpreima  |-  ( Fun 
F  ->  ( `' F " U_ x  e.  A  B )  = 
U_ x  e.  A  ( `' F " B ) )
Distinct variable groups:    x, A    x, F
Allowed substitution hint:    B( x)

Proof of Theorem iunpreima
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eliun 4301 . . . . 5  |-  ( ( F `  y )  e.  U_ x  e.  A  B  <->  E. x  e.  A  ( F `  y )  e.  B
)
21a1i 11 . . . 4  |-  ( Fun 
F  ->  ( ( F `  y )  e.  U_ x  e.  A  B 
<->  E. x  e.  A  ( F `  y )  e.  B ) )
32rabbidv 3072 . . 3  |-  ( Fun 
F  ->  { y  e.  dom  F  |  ( F `  y )  e.  U_ x  e.  A  B }  =  { y  e.  dom  F  |  E. x  e.  A  ( F `  y )  e.  B } )
4 funfn 5627 . . . 4  |-  ( Fun 
F  <->  F  Fn  dom  F )
5 fncnvima2 6016 . . . 4  |-  ( F  Fn  dom  F  -> 
( `' F " U_ x  e.  A  B )  =  {
y  e.  dom  F  |  ( F `  y )  e.  U_ x  e.  A  B } )
64, 5sylbi 198 . . 3  |-  ( Fun 
F  ->  ( `' F " U_ x  e.  A  B )  =  { y  e.  dom  F  |  ( F `  y )  e.  U_ x  e.  A  B } )
7 iunrab 4343 . . . 4  |-  U_ x  e.  A  { y  e.  dom  F  |  ( F `  y )  e.  B }  =  { y  e.  dom  F  |  E. x  e.  A  ( F `  y )  e.  B }
87a1i 11 . . 3  |-  ( Fun 
F  ->  U_ x  e.  A  { y  e. 
dom  F  |  ( F `  y )  e.  B }  =  {
y  e.  dom  F  |  E. x  e.  A  ( F `  y )  e.  B } )
93, 6, 83eqtr4d 2473 . 2  |-  ( Fun 
F  ->  ( `' F " U_ x  e.  A  B )  = 
U_ x  e.  A  { y  e.  dom  F  |  ( F `  y )  e.  B } )
10 fncnvima2 6016 . . . 4  |-  ( F  Fn  dom  F  -> 
( `' F " B )  =  {
y  e.  dom  F  |  ( F `  y )  e.  B } )
114, 10sylbi 198 . . 3  |-  ( Fun 
F  ->  ( `' F " B )  =  { y  e.  dom  F  |  ( F `  y )  e.  B } )
1211iuneq2d 4323 . 2  |-  ( Fun 
F  ->  U_ x  e.  A  ( `' F " B )  =  U_ x  e.  A  {
y  e.  dom  F  |  ( F `  y )  e.  B } )
139, 12eqtr4d 2466 1  |-  ( Fun 
F  ->  ( `' F " U_ x  e.  A  B )  = 
U_ x  e.  A  ( `' F " B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    = wceq 1437    e. wcel 1868   E.wrex 2776   {crab 2779   U_ciun 4296   `'ccnv 4849   dom cdm 4850   "cima 4853   Fun wfun 5592    Fn wfn 5593   ` cfv 5598
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pr 4657
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-fv 5606
This theorem is referenced by: (None)
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