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Theorem funiunfv 6168
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 5640 . . . 4  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
2 funfn 5630 . . . 4  |-  ( Fun  ( F  |`  A )  <-> 
( F  |`  A )  Fn  dom  ( F  |`  A ) )
31, 2sylib 199 . . 3  |-  ( Fun 
F  ->  ( F  |`  A )  Fn  dom  ( F  |`  A ) )
4 fniunfv 6167 . . 3  |-  ( ( F  |`  A )  Fn  dom  ( F  |`  A )  ->  U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )  =  U. ran  ( F  |`  A ) )
53, 4syl 17 . 2  |-  ( Fun 
F  ->  U_ x  e. 
dom  ( F  |`  A ) ( ( F  |`  A ) `  x )  =  U. ran  ( F  |`  A ) )
6 undif2 3873 . . . . 5  |-  ( dom  ( F  |`  A )  u.  ( A  \  dom  ( F  |`  A ) ) )  =  ( dom  ( F  |`  A )  u.  A
)
7 dmres 5144 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
8 inss1 3682 . . . . . . 7  |-  ( A  i^i  dom  F )  C_  A
97, 8eqsstri 3494 . . . . . 6  |-  dom  ( F  |`  A )  C_  A
10 ssequn1 3636 . . . . . 6  |-  ( dom  ( F  |`  A ) 
C_  A  <->  ( dom  ( F  |`  A )  u.  A )  =  A )
119, 10mpbi 211 . . . . 5  |-  ( dom  ( F  |`  A )  u.  A )  =  A
126, 11eqtri 2451 . . . 4  |-  ( dom  ( F  |`  A )  u.  ( A  \  dom  ( F  |`  A ) ) )  =  A
13 iuneq1 4313 . . . 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 4384 . . . 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 3588 . . . . . . . . 9  |-  ( x  e.  ( A  \  dom  ( F  |`  A ) )  ->  -.  x  e.  dom  ( F  |`  A ) )
17 ndmfv 5905 . . . . . . . . 9  |-  ( -.  x  e.  dom  ( F  |`  A )  -> 
( ( F  |`  A ) `  x
)  =  (/) )
1816, 17syl 17 . . . . . . . 8  |-  ( x  e.  ( A  \  dom  ( F  |`  A ) )  ->  ( ( F  |`  A ) `  x )  =  (/) )
1918iuneq2i 4318 . . . . . . 7  |-  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) ( ( F  |`  A ) `  x
)  =  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) (/)
20 iun0 4355 . . . . . . 7  |-  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) (/)  =  (/)
2119, 20eqtri 2451 . . . . . 6  |-  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) ( ( F  |`  A ) `  x
)  =  (/)
2221uneq2i 3617 . . . . 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 3789 . . . . 5  |-  ( U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )  u.  (/) )  = 
U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x
)
2422, 23eqtri 2451 . . . 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 2451 . . 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 5895 . . . 4  |-  ( x  e.  A  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
2726iuneq2i 4318 . . 3  |-  U_ x  e.  A  ( ( F  |`  A ) `  x )  =  U_ x  e.  A  ( F `  x )
2814, 25, 273eqtr3ri 2460 . 2  |-  U_ x  e.  A  ( F `  x )  =  U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )
29 df-ima 4866 . . 3  |-  ( F
" A )  =  ran  ( F  |`  A )
3029unieqi 4228 . 2  |-  U. ( F " A )  = 
U. ran  ( F  |`  A )
315, 28, 303eqtr4g 2488 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 1437    e. wcel 1872    \ cdif 3433    u. cun 3434    i^i cin 3435    C_ wss 3436   (/)c0 3761   U.cuni 4219   U_ciun 4299   dom cdm 4853   ran crn 4854    |` cres 4855   "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-8 1874  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-pow 4602  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-mpt 4484  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:  funiunfvf  6169  eluniima  6170  marypha2lem4  7961  r1limg  8250  r1elssi  8284  r1elss  8285  ackbij2  8680  r1om  8681  ttukeylem6  8951  isacs2  15558  mreacs  15563  acsfn  15564  isacs5  16417  dprdss  17661  dprd2dlem1  17673  dmdprdsplit2lem  17677  uniioombllem3a  22540  uniioombllem4  22542  uniioombllem5  22543  dyadmbl  22556  mblfinlem1  31941  ovoliunnfl  31946  voliunnfl  31948
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