MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  funiunfv Unicode version

Theorem funiunfv 5790
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 5309 . . . 4  |-  ( Fun 
F  ->  Fun  ( F  |`  A ) )
2 funfn 5299 . . . 4  |-  ( Fun  ( F  |`  A )  <-> 
( F  |`  A )  Fn  dom  ( F  |`  A ) )
31, 2sylib 188 . . 3  |-  ( Fun 
F  ->  ( F  |`  A )  Fn  dom  ( F  |`  A ) )
4 fniunfv 5789 . . 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 3543 . . . . 5  |-  ( dom  ( F  |`  A )  u.  ( A  \  dom  ( F  |`  A ) ) )  =  ( dom  ( F  |`  A )  u.  A
)
7 dmres 4992 . . . . . . 7  |-  dom  ( F  |`  A )  =  ( A  i^i  dom  F )
8 inss1 3402 . . . . . . 7  |-  ( A  i^i  dom  F )  C_  A
97, 8eqsstri 3221 . . . . . 6  |-  dom  ( F  |`  A )  C_  A
10 ssequn1 3358 . . . . . 6  |-  ( dom  ( F  |`  A ) 
C_  A  <->  ( dom  ( F  |`  A )  u.  A )  =  A )
119, 10mpbi 199 . . . . 5  |-  ( dom  ( F  |`  A )  u.  A )  =  A
126, 11eqtri 2316 . . . 4  |-  ( dom  ( F  |`  A )  u.  ( A  \  dom  ( F  |`  A ) ) )  =  A
13 iuneq1 3934 . . . 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 3999 . . . 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 3312 . . . . . . . . 9  |-  ( x  e.  ( A  \  dom  ( F  |`  A ) )  ->  -.  x  e.  dom  ( F  |`  A ) )
17 ndmfv 5568 . . . . . . . . 9  |-  ( -.  x  e.  dom  ( F  |`  A )  -> 
( ( F  |`  A ) `  x
)  =  (/) )
1816, 17syl 15 . . . . . . . 8  |-  ( x  e.  ( A  \  dom  ( F  |`  A ) )  ->  ( ( F  |`  A ) `  x )  =  (/) )
1918iuneq2i 3939 . . . . . . 7  |-  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) ( ( F  |`  A ) `  x
)  =  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) (/)
20 iun0 3974 . . . . . . 7  |-  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) (/)  =  (/)
2119, 20eqtri 2316 . . . . . 6  |-  U_ x  e.  ( A  \  dom  ( F  |`  A ) ) ( ( F  |`  A ) `  x
)  =  (/)
2221uneq2i 3339 . . . . 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 3492 . . . . 5  |-  ( U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )  u.  (/) )  = 
U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x
)
2422, 23eqtri 2316 . . . 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 2316 . . 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 5558 . . . 4  |-  ( x  e.  A  ->  (
( F  |`  A ) `
 x )  =  ( F `  x
) )
2726iuneq2i 3939 . . 3  |-  U_ x  e.  A  ( ( F  |`  A ) `  x )  =  U_ x  e.  A  ( F `  x )
2814, 25, 273eqtr3ri 2325 . 2  |-  U_ x  e.  A  ( F `  x )  =  U_ x  e.  dom  ( F  |`  A ) ( ( F  |`  A ) `  x )
29 df-ima 4718 . . 3  |-  ( F
" A )  =  ran  ( F  |`  A )
3029unieqi 3853 . 2  |-  U. ( F " A )  = 
U. ran  ( F  |`  A )
315, 28, 303eqtr4g 2353 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 1632    e. wcel 1696    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   U.cuni 3843   U_ciun 3921   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265    Fn wfn 5266   ` cfv 5271
This theorem is referenced by:  funiunfvf  5791  eluniima  5792  marypha2lem4  7207  r1limg  7459  r1elssi  7493  r1elss  7494  ackbij2  7885  r1om  7886  ttukeylem6  8157  isacs2  13571  mreacs  13576  acsfn  13577  isacs5  14291  dprdss  15280  dprd2dlem1  15292  dmdprdsplit2lem  15296  uniioombllem3a  18955  uniioombllem4  18957  uniioombllem5  18958  dyadmbl  18971  ovoliunnfl  25001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
  Copyright terms: Public domain W3C validator