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Theorem funiunfv 5995
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 5492 . . . 4  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
2 funfn 5482 . . . 4  |-  ( Fun  ( F  |`  A )  <-> 
( F  |`  A )  Fn  dom  ( F  |`  A ) )
31, 2sylib 189 . . 3  |-  ( Fun 
F  ->  ( F  |`  A )  Fn  dom  ( F  |`  A ) )
4 fniunfv 5994 . . 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 3704 . . . . 5  |-  ( dom  ( F  |`  A )  u.  ( A  \  dom  ( F  |`  A ) ) )  =  ( dom  ( F  |`  A )  u.  A
)
7 dmres 5167 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
8 inss1 3561 . . . . . . 7  |-  ( A  i^i  dom  F )  C_  A
97, 8eqsstri 3378 . . . . . 6  |-  dom  ( F  |`  A )  C_  A
10 ssequn1 3517 . . . . . 6  |-  ( dom  ( F  |`  A ) 
C_  A  <->  ( dom  ( F  |`  A )  u.  A )  =  A )
119, 10mpbi 200 . . . . 5  |-  ( dom  ( F  |`  A )  u.  A )  =  A
126, 11eqtri 2456 . . . 4  |-  ( dom  ( F  |`  A )  u.  ( A  \  dom  ( F  |`  A ) ) )  =  A
13 iuneq1 4106 . . . 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 4172 . . . 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 3470 . . . . . . . . 9  |-  ( x  e.  ( A  \  dom  ( F  |`  A ) )  ->  -.  x  e.  dom  ( F  |`  A ) )
17 ndmfv 5755 . . . . . . . . 9  |-  ( -.  x  e.  dom  ( F  |`  A )  -> 
( ( F  |`  A ) `  x
)  =  (/) )
1816, 17syl 16 . . . . . . . 8  |-  ( x  e.  ( A  \  dom  ( F  |`  A ) )  ->  ( ( F  |`  A ) `  x )  =  (/) )
1918iuneq2i 4111 . . . . . . 7  |-  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) ( ( F  |`  A ) `  x
)  =  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) (/)
20 iun0 4147 . . . . . . 7  |-  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) (/)  =  (/)
2119, 20eqtri 2456 . . . . . 6  |-  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) ( ( F  |`  A ) `  x
)  =  (/)
2221uneq2i 3498 . . . . 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 3652 . . . . 5  |-  ( U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )  u.  (/) )  = 
U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x
)
2422, 23eqtri 2456 . . . 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 2456 . . 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 5745 . . . 4  |-  ( x  e.  A  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
2726iuneq2i 4111 . . 3  |-  U_ x  e.  A  ( ( F  |`  A ) `  x )  =  U_ x  e.  A  ( F `  x )
2814, 25, 273eqtr3ri 2465 . 2  |-  U_ x  e.  A  ( F `  x )  =  U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )
29 df-ima 4891 . . 3  |-  ( F
" A )  =  ran  ( F  |`  A )
3029unieqi 4025 . 2  |-  U. ( F " A )  = 
U. ran  ( F  |`  A )
315, 28, 303eqtr4g 2493 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 1652    e. wcel 1725    \ cdif 3317    u. cun 3318    i^i cin 3319    C_ wss 3320   (/)c0 3628   U.cuni 4015   U_ciun 4093   dom cdm 4878   ran crn 4879    |` cres 4880   "cima 4881   Fun wfun 5448    Fn wfn 5449   ` cfv 5454
This theorem is referenced by:  funiunfvf  5996  eluniima  5997  marypha2lem4  7443  r1limg  7697  r1elssi  7731  r1elss  7732  ackbij2  8123  r1om  8124  ttukeylem6  8394  isacs2  13878  mreacs  13883  acsfn  13884  isacs5  14598  dprdss  15587  dprd2dlem1  15599  dmdprdsplit2lem  15603  uniioombllem3a  19476  uniioombllem4  19478  uniioombllem5  19479  dyadmbl  19492  mblfinlem1  26243  ovoliunnfl  26248  voliunnfl  26250
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-fv 5462
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