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Theorem funiunfv 5970
Description: The indexed union of a function's values is the union of its image under the index class.

Note: This theorem depends on the fact that our function value is the empty set outside of its domain. If the antecedent is changed to  F  Fn  A, the theorem can be proved without this dependency. (Contributed by NM, 26-Mar-2006.) (Proof shortened by Mario Carneiro, 31-Aug-2015.)

Assertion
Ref Expression
funiunfv  |-  ( Fun 
F  ->  U_ x  e.  A  ( F `  x )  =  U. ( F " A ) )
Distinct variable groups:    x, A    x, F

Proof of Theorem funiunfv
StepHypRef Expression
1 funres 5462 . . . 4  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
2 funfn 5452 . . . 4  |-  ( Fun  ( F  |`  A )  <-> 
( F  |`  A )  Fn  dom  ( F  |`  A ) )
31, 2sylib 196 . . 3  |-  ( Fun 
F  ->  ( F  |`  A )  Fn  dom  ( F  |`  A ) )
4 fniunfv 5969 . . 3  |-  ( ( F  |`  A )  Fn  dom  ( F  |`  A )  ->  U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )  =  U. ran  ( F  |`  A ) )
53, 4syl 16 . 2  |-  ( Fun 
F  ->  U_ x  e. 
dom  ( F  |`  A ) ( ( F  |`  A ) `  x )  =  U. ran  ( F  |`  A ) )
6 undif2 3760 . . . . 5  |-  ( dom  ( F  |`  A )  u.  ( A  \  dom  ( F  |`  A ) ) )  =  ( dom  ( F  |`  A )  u.  A
)
7 dmres 5136 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
8 inss1 3575 . . . . . . 7  |-  ( A  i^i  dom  F )  C_  A
97, 8eqsstri 3391 . . . . . 6  |-  dom  ( F  |`  A )  C_  A
10 ssequn1 3531 . . . . . 6  |-  ( dom  ( F  |`  A ) 
C_  A  <->  ( dom  ( F  |`  A )  u.  A )  =  A )
119, 10mpbi 208 . . . . 5  |-  ( dom  ( F  |`  A )  u.  A )  =  A
126, 11eqtri 2463 . . . 4  |-  ( dom  ( F  |`  A )  u.  ( A  \  dom  ( F  |`  A ) ) )  =  A
13 iuneq1 4189 . . . 4  |-  ( ( dom  ( F  |`  A )  u.  ( A  \  dom  ( F  |`  A ) ) )  =  A  ->  U_ x  e.  ( dom  ( F  |`  A )  u.  ( A  \  dom  ( F  |`  A ) ) ) ( ( F  |`  A ) `  x
)  =  U_ x  e.  A  ( ( F  |`  A ) `  x ) )
1412, 13ax-mp 5 . . 3  |-  U_ x  e.  ( dom  ( F  |`  A )  u.  ( A  \  dom  ( F  |`  A ) ) ) ( ( F  |`  A ) `  x
)  =  U_ x  e.  A  ( ( F  |`  A ) `  x )
15 iunxun 4257 . . . 4  |-  U_ x  e.  ( dom  ( F  |`  A )  u.  ( A  \  dom  ( F  |`  A ) ) ) ( ( F  |`  A ) `  x
)  =  ( U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )  u.  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) ( ( F  |`  A ) `
 x ) )
16 eldifn 3484 . . . . . . . . 9  |-  ( x  e.  ( A  \  dom  ( F  |`  A ) )  ->  -.  x  e.  dom  ( F  |`  A ) )
17 ndmfv 5719 . . . . . . . . 9  |-  ( -.  x  e.  dom  ( F  |`  A )  -> 
( ( F  |`  A ) `  x
)  =  (/) )
1816, 17syl 16 . . . . . . . 8  |-  ( x  e.  ( A  \  dom  ( F  |`  A ) )  ->  ( ( F  |`  A ) `  x )  =  (/) )
1918iuneq2i 4194 . . . . . . 7  |-  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) ( ( F  |`  A ) `  x
)  =  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) (/)
20 iun0 4231 . . . . . . 7  |-  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) (/)  =  (/)
2119, 20eqtri 2463 . . . . . 6  |-  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) ( ( F  |`  A ) `  x
)  =  (/)
2221uneq2i 3512 . . . . 5  |-  ( U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )  u.  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) ( ( F  |`  A ) `
 x ) )  =  ( U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )  u.  (/) )
23 un0 3667 . . . . 5  |-  ( U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )  u.  (/) )  = 
U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x
)
2422, 23eqtri 2463 . . . 4  |-  ( U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )  u.  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) ( ( F  |`  A ) `
 x ) )  =  U_ x  e. 
dom  ( F  |`  A ) ( ( F  |`  A ) `  x )
2515, 24eqtri 2463 . . 3  |-  U_ x  e.  ( dom  ( F  |`  A )  u.  ( A  \  dom  ( F  |`  A ) ) ) ( ( F  |`  A ) `  x
)  =  U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )
26 fvres 5709 . . . 4  |-  ( x  e.  A  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
2726iuneq2i 4194 . . 3  |-  U_ x  e.  A  ( ( F  |`  A ) `  x )  =  U_ x  e.  A  ( F `  x )
2814, 25, 273eqtr3ri 2472 . 2  |-  U_ x  e.  A  ( F `  x )  =  U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )
29 df-ima 4858 . . 3  |-  ( F
" A )  =  ran  ( F  |`  A )
3029unieqi 4105 . 2  |-  U. ( F " A )  = 
U. ran  ( F  |`  A )
315, 28, 303eqtr4g 2500 1  |-  ( Fun 
F  ->  U_ x  e.  A  ( F `  x )  =  U. ( F " A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1369    e. wcel 1756    \ cdif 3330    u. cun 3331    i^i cin 3332    C_ wss 3333   (/)c0 3642   U.cuni 4096   U_ciun 4176   dom cdm 4845   ran crn 4846    |` cres 4847   "cima 4848   Fun wfun 5417    Fn wfn 5418   ` cfv 5423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-fv 5431
This theorem is referenced by:  funiunfvf  5971  eluniima  5972  marypha2lem4  7693  r1limg  7983  r1elssi  8017  r1elss  8018  ackbij2  8417  r1om  8418  ttukeylem6  8688  isacs2  14596  mreacs  14601  acsfn  14602  isacs5  15347  dprdss  16531  dprd2dlem1  16545  dmdprdsplit2lem  16549  uniioombllem3a  21069  uniioombllem4  21071  uniioombllem5  21072  dyadmbl  21085  mblfinlem1  28433  ovoliunnfl  28438  voliunnfl  28440
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