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

Theorem elunirn 6149
Description: Membership in the union of the range of a function. See elunirnALT 6150 for a shorter proof which uses ax-pow 4625. (Contributed by NM, 24-Sep-2006.)
Assertion
Ref Expression
elunirn  |-  ( Fun 
F  ->  ( A  e.  U. ran  F  <->  E. x  e.  dom  F  A  e.  ( F `  x
) ) )
Distinct variable groups:    x, A    x, F

Proof of Theorem elunirn
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 eluni 4248 . 2  |-  ( A  e.  U. ran  F  <->  E. y ( A  e.  y  /\  y  e. 
ran  F ) )
2 funfn 5615 . . . . . . . 8  |-  ( Fun 
F  <->  F  Fn  dom  F )
3 fvelrnb 5913 . . . . . . . 8  |-  ( F  Fn  dom  F  -> 
( y  e.  ran  F  <->  E. x  e.  dom  F ( F `  x
)  =  y ) )
42, 3sylbi 195 . . . . . . 7  |-  ( Fun 
F  ->  ( y  e.  ran  F  <->  E. x  e.  dom  F ( F `
 x )  =  y ) )
54anbi2d 703 . . . . . 6  |-  ( Fun 
F  ->  ( ( A  e.  y  /\  y  e.  ran  F )  <-> 
( A  e.  y  /\  E. x  e. 
dom  F ( F `
 x )  =  y ) ) )
6 r19.42v 3016 . . . . . 6  |-  ( E. x  e.  dom  F
( A  e.  y  /\  ( F `  x )  =  y )  <->  ( A  e.  y  /\  E. x  e.  dom  F ( F `
 x )  =  y ) )
75, 6syl6bbr 263 . . . . 5  |-  ( Fun 
F  ->  ( ( A  e.  y  /\  y  e.  ran  F )  <->  E. x  e.  dom  F ( A  e.  y  /\  ( F `  x )  =  y ) ) )
8 eleq2 2540 . . . . . . 7  |-  ( ( F `  x )  =  y  ->  ( A  e.  ( F `  x )  <->  A  e.  y ) )
98biimparc 487 . . . . . 6  |-  ( ( A  e.  y  /\  ( F `  x )  =  y )  ->  A  e.  ( F `  x ) )
109reximi 2932 . . . . 5  |-  ( E. x  e.  dom  F
( A  e.  y  /\  ( F `  x )  =  y )  ->  E. x  e.  dom  F  A  e.  ( F `  x
) )
117, 10syl6bi 228 . . . 4  |-  ( Fun 
F  ->  ( ( A  e.  y  /\  y  e.  ran  F )  ->  E. x  e.  dom  F  A  e.  ( F `
 x ) ) )
1211exlimdv 1700 . . 3  |-  ( Fun 
F  ->  ( E. y ( A  e.  y  /\  y  e. 
ran  F )  ->  E. x  e.  dom  F  A  e.  ( F `
 x ) ) )
13 fvelrn 6015 . . . . . . 7  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( F `  x
)  e.  ran  F
)
1413a1d 25 . . . . . 6  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( A  e.  ( F `  x )  ->  ( F `  x )  e.  ran  F ) )
1514ancld 553 . . . . 5  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( A  e.  ( F `  x )  ->  ( A  e.  ( F `  x
)  /\  ( F `  x )  e.  ran  F ) ) )
16 fvex 5874 . . . . . 6  |-  ( F `
 x )  e. 
_V
17 eleq2 2540 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  ( A  e.  y  <->  A  e.  ( F `  x ) ) )
18 eleq1 2539 . . . . . . 7  |-  ( y  =  ( F `  x )  ->  (
y  e.  ran  F  <->  ( F `  x )  e.  ran  F ) )
1917, 18anbi12d 710 . . . . . 6  |-  ( y  =  ( F `  x )  ->  (
( A  e.  y  /\  y  e.  ran  F )  <->  ( A  e.  ( F `  x
)  /\  ( F `  x )  e.  ran  F ) ) )
2016, 19spcev 3205 . . . . 5  |-  ( ( A  e.  ( F `
 x )  /\  ( F `  x )  e.  ran  F )  ->  E. y ( A  e.  y  /\  y  e.  ran  F ) )
2115, 20syl6 33 . . . 4  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( A  e.  ( F `  x )  ->  E. y ( A  e.  y  /\  y  e.  ran  F ) ) )
2221rexlimdva 2955 . . 3  |-  ( Fun 
F  ->  ( E. x  e.  dom  F  A  e.  ( F `  x
)  ->  E. y
( A  e.  y  /\  y  e.  ran  F ) ) )
2312, 22impbid 191 . 2  |-  ( Fun 
F  ->  ( E. y ( A  e.  y  /\  y  e. 
ran  F )  <->  E. x  e.  dom  F  A  e.  ( F `  x
) ) )
241, 23syl5bb 257 1  |-  ( Fun 
F  ->  ( A  e.  U. ran  F  <->  E. x  e.  dom  F  A  e.  ( F `  x
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   E.wrex 2815   U.cuni 4245   dom cdm 4999   ran crn 5000   Fun wfun 5580    Fn wfn 5581   ` cfv 5586
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594
This theorem is referenced by:  fnunirn  6151  fin23lem30  8718  ustn0  20458  elrnust  20462  ustbas  20465  metuvalOLD  20787  metuval  20788  elunirn2  27161  metidval  27505  pstmval  27510  elunirnmbfm  27864  fourierdlem70  31477  fourierdlem71  31478  fourierdlem80  31487
  Copyright terms: Public domain W3C validator