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Theorem funiunfv 5774
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 5293 . . . 4  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
2 funfn 5283 . . . 4  |-  ( Fun  ( F  |`  A )  <-> 
( F  |`  A )  Fn  dom  ( F  |`  A ) )
31, 2sylib 188 . . 3  |-  ( Fun 
F  ->  ( F  |`  A )  Fn  dom  ( F  |`  A ) )
4 fniunfv 5773 . . 3  |-  ( ( F  |`  A )  Fn  dom  ( F  |`  A )  ->  U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )  =  U. ran  ( F  |`  A ) )
53, 4syl 15 . 2  |-  ( Fun 
F  ->  U_ x  e. 
dom  ( F  |`  A ) ( ( F  |`  A ) `  x )  =  U. ran  ( F  |`  A ) )
6 undif2 3530 . . . . 5  |-  ( dom  ( F  |`  A )  u.  ( A  \  dom  ( F  |`  A ) ) )  =  ( dom  ( F  |`  A )  u.  A
)
7 dmres 4976 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
8 inss1 3389 . . . . . . 7  |-  ( A  i^i  dom  F )  C_  A
97, 8eqsstri 3208 . . . . . 6  |-  dom  ( F  |`  A )  C_  A
10 ssequn1 3345 . . . . . 6  |-  ( dom  ( F  |`  A ) 
C_  A  <->  ( dom  ( F  |`  A )  u.  A )  =  A )
119, 10mpbi 199 . . . . 5  |-  ( dom  ( F  |`  A )  u.  A )  =  A
126, 11eqtri 2303 . . . 4  |-  ( dom  ( F  |`  A )  u.  ( A  \  dom  ( F  |`  A ) ) )  =  A
13 iuneq1 3918 . . . 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 8 . . 3  |-  U_ x  e.  ( dom  ( F  |`  A )  u.  ( A  \  dom  ( F  |`  A ) ) ) ( ( F  |`  A ) `  x
)  =  U_ x  e.  A  ( ( F  |`  A ) `  x )
15 iunxun 3983 . . . 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 3299 . . . . . . . . 9  |-  ( x  e.  ( A  \  dom  ( F  |`  A ) )  ->  -.  x  e.  dom  ( F  |`  A ) )
17 ndmfv 5552 . . . . . . . . 9  |-  ( -.  x  e.  dom  ( F  |`  A )  -> 
( ( F  |`  A ) `  x
)  =  (/) )
1816, 17syl 15 . . . . . . . 8  |-  ( x  e.  ( A  \  dom  ( F  |`  A ) )  ->  ( ( F  |`  A ) `  x )  =  (/) )
1918iuneq2i 3923 . . . . . . 7  |-  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) ( ( F  |`  A ) `  x
)  =  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) (/)
20 iun0 3958 . . . . . . 7  |-  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) (/)  =  (/)
2119, 20eqtri 2303 . . . . . 6  |-  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) ( ( F  |`  A ) `  x
)  =  (/)
2221uneq2i 3326 . . . . 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 3479 . . . . 5  |-  ( U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )  u.  (/) )  = 
U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x
)
2422, 23eqtri 2303 . . . 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 2303 . . 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 5542 . . . 4  |-  ( x  e.  A  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
2726iuneq2i 3923 . . 3  |-  U_ x  e.  A  ( ( F  |`  A ) `  x )  =  U_ x  e.  A  ( F `  x )
2814, 25, 273eqtr3ri 2312 . 2  |-  U_ x  e.  A  ( F `  x )  =  U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )
29 df-ima 4702 . . 3  |-  ( F
" A )  =  ran  ( F  |`  A )
3029unieqi 3837 . 2  |-  U. ( F " A )  = 
U. ran  ( F  |`  A )
315, 28, 303eqtr4g 2340 1  |-  ( Fun 
F  ->  U_ x  e.  A  ( F `  x )  =  U. ( F " A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684    \ cdif 3149    u. cun 3150    i^i cin 3151    C_ wss 3152   (/)c0 3455   U.cuni 3827   U_ciun 3905   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  funiunfvf  5775  eluniima  5776  marypha2lem4  7191  r1limg  7443  r1elssi  7477  r1elss  7478  ackbij2  7869  r1om  7870  ttukeylem6  8141  isacs2  13555  mreacs  13560  acsfn  13561  isacs5  14275  dprdss  15264  dprd2dlem1  15276  dmdprdsplit2lem  15280  uniioombllem3a  18939  uniioombllem4  18941  uniioombllem5  18942  dyadmbl  18955
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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